Pierre Lochak
email: pierre.lochak@imj-prg.fr

ON PROCONGRUENCE CURVE COMPLEXES AND THEIR AUTOMORPHISMS

Keywords: AMS Math Classification: 11R32, 14D22, 57M99

). We work all along with the procongruence completions -and for good reasons -recalling however that the so-called congruence conjecture predicts that this completion should coincide with the full profinite completion.

Introduction

The primary goal of this paper is to start laying the foundations for a topological version of Grothendieck-Teichmüller theory and the goal of this short introduction is to provide some clues as to what this could mean ; and of course about the contents of the paper. For much more on the background landscape we refer once and for all to [START_REF] Lochak | Results and conjectures in profinite Teichmüller theory[END_REF] and its references. Because numerous objects are involved we have gathered the main (essentially classical) definitions in a short Appendix which the reader is invited to consult when (s)he feels like it. We will also explicitly refer to it.

In a few words which will be considerably expanded below and possibly elsewhere, the situation can be described as follows. Let S = S g,n be a hyperbolic surface of finite type (cf. §A.1); it has (modular) dimension d(S) = 3g -3 + n which can be seen for instance as the (complex) dimension of the modular orbifold M(S) (cf. §A.2) or else as the maximal number of non intersecting simple closed curves lying on S, considered up to isotopy (these objects form a set which we denote L(S)). Starting from L(S) one builds several (simplicial, non locally finite) complexes, especially the curve complex C(S) (cf. §A.5), of dimension d(S) -1, and the so-called two-dimensional pants complex C P (S) (cf. §A.7) of which it is enough to consider the 1-skeleton (the pants graph). The attached Teichmüller group (a.k.a. mapping class group) Γ(S) (cf. §A.3) acts naturally on these objects (L(S), C(S), C P (S)).

The curve complex C(S) was first constructed by W.J.Harvey in close analogy with buildings for reductive groups, from which the significance of its automorphisms was immediately recognized (see [START_REF] Lochak | Results and conjectures in profinite Teichmüller theory[END_REF], Introduction, for a more detailed story and references). It was shown in the eighties, by N.V.Ivanov (cf. [START_REF] Ivanov | Subgroups of Teichmüller modular groups[END_REF]) and J.L.Harer (cf. [START_REF] Harer | Stability of the homology of the mapping class groups of orientable surfaces[END_REF][START_REF] Harer | The cohomology of the moduli spaces of curves[END_REF]) independently, that the curve complex C(S) has the homotopy type of a wedge of spheres, an important and fundationnal result. A few years later N.V.Ivanov proved (cf. [START_REF] Ivanov | Automorphisms of complexes of curves and of Teichmüller spaces[END_REF] as well as [START_REF] Luo | Automorphisms of the complex of curves[END_REF]) that C(S) is essentially rigid, the only automorphism not arising from the action of Γ(S) being the mirror reflection (an orientation reversing automorphism of the underlying surface). This is embodied in the exact sequence (A 2) of §A.12. An important point is that it also enables one to control the automorphisms of the group Γ(S), leading to the exact sequence (A 3), and indeed the automorphisms of any cofinite subgroup Γ λ (S) ⊂ Γ(S). The upshot is thus that both C(S) and Γ(S) are rigid with the mirror reflection as only non inner automorphism; in anticipation one can identify the reflection with complex conjugacy and consider that it generates the Galois group Gal(C/R) Z/2.

The pants complex C P (S) was defined somewhat later and its automorphisms were considered relatively recently. D.Margalit showed (in [START_REF] Margalit | Automorphisms of the pants complex[END_REF]) that it is rigid as well, more precisely that one can replace C(S) by C P (S) in the sequence (A 2), so that Aut(C P (S)) = Aut(C(S)). This result will be reproved below (in §2) in a different way. Now to completions; they were introduced in [START_REF] Boggi | Profinite Teichmüller theory[END_REF] in an effort to attack the congruence conjecture (cf. §A.10). Although this was actually not achieved in [START_REF] Boggi | Profinite Teichmüller theory[END_REF] (see the review of D.Abramovitch in MathSciNet for a careful and well intended discussion), the idea of completing various geometric or in fact topological objects (cf. §A.11), primarily versions of the complexes of curves, appears as a deep and potentially fruitful one. Perhaps the main point or slogan of the present paper is that the automorphisms of the completed complexes have a lot to do with Grothendieck-Teichmüller theory (in all genera, not only genus 0) and the corresponding group. This is also the main theme of the manuscript [START_REF] Lochak | Automorphism groups of profinite complexes of curves and the Grothendieck-Teichmüller group[END_REF] (2007, unpublished).

More specifically let Ĉ(S) and ĈP (S) denote the respective profinite completions of the curves and pants complexes. Then ĈP (S) remains rigid whereas Ĉ(S) acquires an enormous automorphism group, which is precisely (a somewhat sophisticated version of) the Grothendieck-Teichmüller group. These issues are discussed in detail in [START_REF] Lochak | Automorphism groups of profinite complexes of curves and the Grothendieck-Teichmüller group[END_REF] but watertight proofs are missing there, for technical reasons which in some sense amount to the fact that one does not know how to prove (the highly plausible fact) that Ĉ(S) is isomorphic to the profinite completion ĈG (S) of the group theoretic version C G (S) of the curve complex (cf. A.6).

Fortunately things become somewhat easier when working with the congruence completions. In terms of covers the congruence completion Γ(S) describes the (orbifold unramified finite) covers of the modular orbifold M(S) arising from covers of S itself, which are obviously much more manageable. Whether these covers are cofinal or not is the question which the congruence conjecture purports to answer in a positive way. In any event here we take advantage of the results shown in particular in [START_REF] Boggi | On the procongruence completion of the Teichmüller modular groups[END_REF] to attack the questions in the framework of the congruence completions. Turning to the procongruence complexes Č(S) and ČP (S) we prove that the latter one, namely the procongruence pants complex, remains rigid. That is we have a short exact sequence:

1 → Inn( Γ(S)) → Aut(( ČP (S)

) st ) → Z/2 → 1.
This is a somewhat cryptic and incomplete version of the result. The subscript st stand for "stack" and the exact definition of these objects, which involves the so-called topological stacks, will be detailed in due time, at the beginning of section 7. One can again state that (with the mild exception of type (1, 2)) Out(( ČP (S)) st ) Z/2 Gal(C/R), just as in the discrete case, and the nontrivial outer automorphism comes again from orientation or complex conjugacy.

At first sight this may appear as a rather dull result: the procongruence pants complex is rigid, and this is also the case in the full profinite setting, modulo the congruence conjecture. In other words, rigidity survives completion in that case. So what? The point is that there are at least one surprise and one application in store. The surprise -if any -consists in the fact that the procongruence curve complex is not rigid. Far from it; indeed the outer automorphism group Out( Č(S)) (for d(S) > 3, say) is enormous and can be taken as a higher genus version of the Grothendieck-Teichmüller group. In particular it is independent of S, that is of the type (g, n), and it naturally contains the absolute Galois group Gal( Q/Q). This and much more is elaborated in [START_REF] Lochak | Automorphism groups of profinite complexes of curves and the Grothendieck-Teichmüller group[END_REF] (see also [START_REF] Lochak | Results and conjectures in profinite Teichmüller theory[END_REF]) which however again does not contain watertight proofs inasmuch as the setting there is that of full profinite completion where certain tools are still lacking, in contrast with the case of the procongruence completion. The upshot is that the rigidity result shown in the present paper should pave the way for a thorough investigation of this new, topological version of Grothendieck-Teichmüller theory.

To end this introduction in a concrete fashion, we give a tour of the paper, highlighting some of the key concepts and statements. Most definitions are to be found in the Appendix; a few appear in the body of the text. Section 2 deals with three types of curve complexes attached to the surface S, namely C(S), C P (S) and C * (S). The setting is the classical discrete one ; the main, important and often difficult classical results are recalled in §2.1 and at the end of the Appendix. Yet even in this discrete framework one will notice two novel features: First, the emphasis on the role of the graph C * (S) (defined in §A.8), second the very idea of reconstruction in §2.2. Although the paper is geared towards treating the case of complete (more accurately, completed) complexes, reconstruction, as embodied in Theorem 2.10, already bears new fruits in the discrete setting, recovering in particular the discrete rigidity Theorem 2.13, which features the main result of [START_REF] Margalit | Automorphisms of the pants complex[END_REF]. Section 3 introduces completions, which in this setting are quite a new notion, introduced by M. Boggi a few years ago and never discussed anywhere outside of his own papers. This is why we found it necessary to include some easy but not trivial and often somewhat counterintuitive properties. The reader may recall how "exotic" it may appear to pass from the ordinary integers (Z) to the p-adic ones (Z p ). The situation here is analogous, only substantially more intricate. The main result of this section is the isomorphism of three versions of the curve complex, as stated in Theorem 3.1. Note that this same isomorphism in the discrete case is elementary. Although this basic and important result is stated in [START_REF] Boggi | On the procongruence completion of the Teichmüller modular groups[END_REF], the proof there does not appear to be completely satisfactory, so we present a new, hopefully waterproof, one. As a general remark, one should stress that we are treading fairly new and slippery ground, say the profinite geometry of surfaces, so that it seems useful, indeed necessary, to do one's best in order to make it firmer and sometimes add in a little more context, like e.g. in §3.3 as well as in several "remarks" along the text. Section 4 provides part of the necessary and quite nontrivial dictionary between graphs (more generally complexes) and the more traditional group theoretic setting. These translations are often elementary in the discrete setting but not always. For instance the discrete analog of Proposition 4.4 uses some basic results from Thurston's theory of diffeomorphisms of surfaces, something which has no analog in the complete case (see [START_REF] Lochak | Results and conjectures in profinite Teichmüller theory[END_REF]). This section owes a lot to [START_REF] Boggi | On the procongruence completion of the Teichmüller modular groups[END_REF] and discussions with its author. Yet it is self-contained and includes several new results and/or proofs. Proposition 4.4 and Theorem 4.5 are particularly noticeable. Section 5 starts the exploration of the congruence case, recalling that the congruence conjecture precisely predicts that it coincides with the full profinite one. We refer to the introduction of that section for details about its content. Theorem 5.1 is new, and so is of course the reconstruction result (Theorem 5.13) in the complete case, as well as Proposition 5.15. It should be stressed that some "easy" results (God given, so to speak) are in fact of the utmost importance here. This is indeed the case of Proposition 5.8 (see also Remark 5.2 beneath that result), which states (and proves) the existence of an important stratification of the moduli stacks of curves. This property is crucial and not (yet) available in the full profinite setting. In fact its validity is equivalent to that of the congruence conjecture, as explained in Remark 5.2 ii). Section 6 takes up the question of the automorphisms of the congruence completed complexes, probably the main goal of the paper, on the road to a topological version of Grothendieck-Teichmüller theory, true to Grothendieck's watermark indications in his Esquisse d'un programme. We refer again to the introduction of that section for more detail and context. The content of that section is essentially new, building in particular on the reconstruction results of §5.2. For instance Propositions 6.1, 6.2 and 6.3 appear as easy but important corollaries of these results. The property of type preservation for automorphisms of the procongruence curve complex (Theorem 6.4) is quite significant and its proof cannot really mimic the topologically inspired proof in the discrete case. In §6.2 we translate again the situation from group automorphisms to automorphisms of (complete) complexes. Yet, as examplified in Proposition 6.8, an important property emerges, namely that automorphisms of complexes correspond to virtual group automorphisms, that is automorphisms up to passing to finite index subgroups (étale covers, in geometric terms for fundamental groups). Note that this property in turn relies crucially on Proposition 4.4, describing a certain lattice property of the (open subgroups of the) free abelian groups generated by commuting (pro)twists. The short paragraph §6.3 finally introduces the arithmetic Galois group G Q and shows that it acts naturally and faithfully on the congruence completed curve complex (Proposition 6.9). The same property holds true for a variant of the profinite Grothendieck-Teichmüller group constructed by the author together with H.Nakamura and L.Schneps some years ago (Proposition 6.10). Note that this version, denoted IΓ, is rather more sophisticated than the one which appears in modern deformation theory (denoted GT (Q) after Drinfel'd) as it is profinite (as opposed to prounipotent) and adapted to every genus (as opposed to genus 0 only). In particular it captures the whole of G Q (see below Proposition 6.10), as opposed to the quotient defined by the maximal nilpotent extension of Q. However, all in all, §6.3 should be considered a "prequel" to a much more extensive investigation (with a somewhat preliminary version appearing in [START_REF] Lochak | Automorphism groups of profinite complexes of curves and the Grothendieck-Teichmüller group[END_REF]). The final Section 7 is essentially devoted to an important fundational result stated as Theorem 7.1. Here the main message is that this result is dramatically wrong if one replaces ČP (S) (assuming congruence completion for definiteness ; the result is more general as the statement indicates) with Č(S). Not only the curve complex is not rigid but the deformation group is precisely the (enormous) Grothendieck-Teichmüller group (much more can be gathered from the text as well as [START_REF] Lochak | Automorphism groups of profinite complexes of curves and the Grothendieck-Teichmüller group[END_REF] and [START_REF] Lochak | Results and conjectures in profinite Teichmüller theory[END_REF]). The proof, indeed the precise statement, requires an excursion into B.Noohi's topological stacks ( §7.2). It also crucially uses one of the main results of [START_REF] Hatcher | On the Teichmüller tower of mapping class groups[END_REF]. We have added in §7.5 a more detailed exposition of the one-dimensional case (which covers the case of "dessins d'enfant") and finally a geometric, more precisely modular, interpretation of some of the notions and results connected with C P (S) and its completion ( §7.6). Acknowledgments. This paper benefited from conversations and exchanges with M. Boggi and L. Funar, who after the near completion of the present text decided to produce a different exposition of part of this material. The attentive reader will find that the differences actually overshadow some obvious similarities.

Discrete complexes : rigidity and reconstruction

In this section we prepare the ground by recalling some rigidity results in the discrete setting in a fashion taylored to our needs (see §A.12 for a tightly compressed summary) and prove a reconstruction result which later on will be adapted to the procongruence setting; as a side benefit it provides another proof of the main result of [START_REF] Margalit | Automorphisms of the pants complex[END_REF], that is the rigidity of the discrete pants complex. To a hyperbolic surface S we associate in particular three graphs, namely the 1-skeleton C (1) (S) of the curve complex (cf. §A.5), the pants graph C

(1) P (S) (cf. §A.7) and the graph C * (S) (cf. §A.8). The definitions readily extend (cf. §A.9) to the case of not necessarily connected surfaces, with hyperbolic connected. These three graphs, and later on their respective completions, carry all the information we need. In some sense we are trying to pass from an essentially group theoretic framework, revolving around the Teichmüller group Γ(S) (cf. §A.3), its completions and their cofinite subgroups to a graph theoretic one, based on the graphs above and later their completions, together with certain subgraphs.

2.1. Rigidity of the discrete curves complex. Basically this paragraph revolves around the two short exact sequences of §A.12. We start with the curve complex C(S) and consider its group of simplicial automorphisms Aut(C(S)). There is a natural map M od(S) → Aut(C(S)) induced by letting a diffeomorphism act on loops (i.e. elements of L(S) = C (0) (S); cf. §A.5), everything up to isotopy. The elements of the center of the left-hand group lie in the kernel of that map because they commute with twists, so there is an induced map θ : Inn(M od(S))) → Aut(C(S)). Assume now that C(S) is connected, that is d(S) > 1. Then it is not too difficult to show that θ is injective. The deep fundamental fact mentioned in the introduction and embodied by the sequence (A 2) states that θ is also surjective for (g, n) = (1, 2). This surjectivity, in item i) below, is due to N.V. Ivanov ([16]) and F. Luo ([18]): Theorem 2.1. Let S be a connected hyperbolic surface of type (g, n) with d(S) > 1. Then i) the natural map θ :

Inn(M od(S)) → Aut(C(S)) is an isomorphism except if (g, n) = (1, 2)
, in which case it is injective but not surjective; in fact θ maps Inn(M od(S 1,2 )) onto the strict subgroup of the elements Aut(C(S 1,2 )) which globally preserve the set of vertices representing nonseparating curves; ii) Aut(C (1)

(S)) = Aut(C(S)).
Of course, if the type is different from (1, 2) and (2, 0), M od(S) is centerfree and θ provides an isomorphism between M od(S) and Aut(C(S)). Item ii) is easy but quite telling; it confirms that the pants complex and the pants graph (i.e. its 1-skeleton) have the same automorphisms. This fact will remain valid after completion. Here is a short proof. There is a natural map Aut(C(S)) → Aut(C (1) (S)) which is injective; indeed the restriction to the set of vertices is already injective. To prove surjectivity it is enough to give a graph theoretic characterization of the higher dimensional simplices of C(S) and this is easily available: a moment contemplation will confirm that the k-dimensional simplices are in one-to-one correspondence with the complete subgraphs (a.k.a. cliques) of C (1) (S) with k + 1 vertices, i.e. subgraphs such that any two vertices are connected by an edge. This characterization proves ii). Note that to any simplicial complex one can associate the complex obtained by adding in all the cliques as simplices. Here C(S) is a flag complex, that is, its simplices are exactly given by the cliques. This will also be the case of the other complexes we will meet (including in the profinite world) and it says that in fact all the information is contained is contained in a graph, namely the 1-skeleton of the relevant complex.

Remark 2.1. The odd looking case of type (1, 2) is actually easy to understand. It stems from the fact that C(S 1,2 ) and C(S 0,5 ) are isomorphic, whereas Γ 1, [START_REF] Artin | Etale Homotopy[END_REF] /Z(Γ 1, [START_REF] Artin | Etale Homotopy[END_REF] ) maps into Γ 0, [START_REF] Boggi | Galois coverings of moduli spaces of curves and loci of curves with symmetry[END_REF] as a subgroup of index 5; indeed θ maps Inn(M od(S 1,2 )) injectively onto an index 5 subgroup of Aut(C(S 1,2 )). See §A.4 and [START_REF] Lochak | Automorphism groups of profinite complexes of curves and the Grothendieck-Teichmüller group[END_REF][START_REF] Margalit | Automorphisms of the pants complex[END_REF] for a geometric discussion.

N.V.Ivanov went on to show how to use the description of Aut(C(S)) afforded by Theorem 2.1 in order to study the action of Γ(S) on Teichmüller space. He recovered in this way ( [START_REF] Ivanov | Automorphisms of complexes of curves and of Teichmüller spaces[END_REF]) the classical result of H.Royden about automorphisms of Teichmüller spaces: Corollary 2.2. If d(S) > 1, any complex automorphism of T (S) is induced by an element of M od(S).

As N.V.Ivanov again showed, Theorem 2.1 also has immediate bearing on the automorphisms of the modular groups. Here we require one more definition, which will turn out to be of typical anabelian flavor: Definition 2.3. An element of Aut(Γ(S)) is called inertia preserving if it (globally) preserves the set of cyclic subgroups generated by Dehn twists, that is maps a twist in Γ(S) to a power of some other twist.

For a geometric discussion justifying this terminology we refer e.g. to [START_REF] Lochak | Results and conjectures in profinite Teichmüller theory[END_REF]. In the present discrete setting we have the following This result, which again is essentially due to N.V.Ivanov (cf. [START_REF] Ivanov | Subgroups of Teichmüller modular groups[END_REF] and references therein) rests on a group theoretic characterization of twists inside Γ(S). It is rarely stated independently or emphasized but we would like to stress it in view of the profinite or procongruence case; we also refer to [START_REF] Mccarthy | Automorphisms of surface mapping class groups: A recent theorem of N.Ivanov[END_REF] for a nice proof based on the notion of stable rank. This is because first we do not know how to prove the profinite or procongruence analog, which is unfortunate, and second because in the profinite setting this would feature a rather striking and precise analog of the so-called "local correspondence" in birational anabelian geometry (see [START_REF] Lochak | Results and conjectures in profinite Teichmüller theory[END_REF] for more detail). Armed with Theorem 2.4 it is easy to use Theorem 2.1 in order to study the automorphisms of Γ(S). Actually it turns out to be no more difficult to study morphisms between all the cofinite subgroups, (cf. [START_REF] Ivanov | Automorphisms of complexes of curves and of Teichmüller spaces[END_REF], Theorem 2); we state this as: Corollary 2.5. Assume d(S) > 1 and Γ = Γ(S) has trivial center; let Γ 1 , Γ 2 ⊂ Γ be two finite index subgroups. Then any isomorphism φ between Γ 1 and Γ 2 is induced by an element of M od(S), namely there exists g ∈ M od(S) such that φ(g 1 ) = g -1 g 1 g for any g 1 ∈ Γ 1 . In particular Out(Γ(S)) Z/2.

As usual one can study the two cases with nontrivial center, that is (1, 2) and (2, 0) in detail; see [START_REF] Mccarthy | Automorphisms of surface mapping class groups: A recent theorem of N.Ivanov[END_REF] for the latter one. This ends our review of the rigidity properties of the curves complex in the discrete setting, together with the group theoretic consequences. Before switching to the pants complex (or graph), we now introduce a kind of reconstruction technique for the various complexes.

2.2.

Reconstructing complexes and the rigidity of the pants graph. In this paragraph we explore the local structure of our three complexes C(S), C * (S) and C P (S) and show how to reconstruct them from local data. We especially focus on the three graphs obtained by retaining only the 1-skeleta of C(S) and C P (S). As mentioned already we often abuse notation by writing C P (S) for the pants graph, bearing in mind that the full two-dimensional complex can be reconstructed from its 1-skeleton (cf. [START_REF] Margalit | Automorphisms of the pants complex[END_REF]). Trivially we have C P (S) → C * (S); it will turn out that this inclusion or rather its (equally trivial) analog after completion is of fundamental importance and lies in some sense at the very basis of a topological version of Grothendieck-Teichmüller theory. Note that (for d(S) > 1) C * (S) is the 1-skeleton of the dual of the simplicial complex C(S). In terms of automorphisms C * (S) carries essentially the same information as C(S) (see below for a precise statement) and it has been introduced essentially with a view to the above inclusion. Here we show (in the discrete setting) how to reconstruct the complexes from local data. Rigidity of the pants graph and a fortiori of the full complex will appear as an easy corollary. The proof of the reconstruction result (Theorem 2.10) is given in the next subsection.

Let us move to concrete and elementary notions. Given a surface S, a subsurface T is defined as T = S \ σ where σ ∈ C(S). We denote it S σ ; it is nothing but S cut or slit along the multicurve representing σ. In this definition the curves are defined as usual up to isotopy and one can choose a representative of the multicurve. One way to do this in a coherent way is to equip S with a (any) metric of constant negative curvature and use the (unique) geodesic representatives of the various multicurves. The metric plainly induces a metric with the same property on all the subsurfaces of S. There is a natural inclusion C * (S σ ) ⊂ C * (S); in fact C * (S σ ) is the full subgraph of C * (S) whose vertices correspond to those pants decompositions of S which include σ (ditto for C P (S)). For σ ∈ C(S), we let |σ| denote the number of curves which constitute σ. So |σ| = dim(σ) + 1 if dim(σ) denotes the dimension of the simplex σ ∈ C(S). The quantity |σ| turns out to be more convenient in our context; in particular d(S σ ) = d(S) -|σ|. We include throughout the case of an empty cell (dimension -1): S ∅ = S. For example if σ is a maximal multicurve (pants decomposition), S σ is a disjoint union of pants and C * (S σ ) is empty or reduced to a point (cf. §A.8) depending on convention. We call two simplices ρ, σ ∈ C(S) compatible if the curves which compose ρ and σ do not intersect properly, that is they are either disjoint or coincide. Complex theoretically it means that ρ and σ lie in the closure of a common top dimensional simplex of C(S). If ρ and σ are compatible, we define their unions and intersections ρ ∪ σ, ρ ∩ σ ∈ C(S) in the obvious way. Then we clearly have: A maximal copy of F is a subgraph of C P (S) which is isomorphic to F and is not properly contained in another such subgraph. Note that the operation described in this lemma is not reversible; one cannot recognize C P (S) inside C * (S) without additional information and this may well be the seed of Grothendieck-Teichmüller theory. For the time being we note the following consequence in terms of automorphisms: Lemma 2.8.

Aut(C P (S)) ⊂ Aut(C * (S))

Proof. An automorphism of C P (S) determines a permutation of the common vertex set V (S) (cf. §A.9), which in turn defines an automorphism of C * (S) provided it is compatible with its edges. Lemma 2.7 and the fact that G is a complete graph ensure that this is always the case.

So any automorphism of C P (S) determines an automorphism of C * (S) because both graphs share the same set of vertices and automorphisms of complexes are determined by their effect on the vertices. However a priori only certain automorphisms of C * (S) will preserve the additional structure given by the edges of C P (S), inducing an automorphism of this subgraph. In dimension 1, Aut(G) is nothing but the permutation group on its vertices. Any automorphism of F determines a unique automorphism of G by looking at its effect on the vertices, but Aut(F ) P GL 2 (Z) is certainly much smaller than Aut(G). In the discrete case a kind of rigidification occurs for d(S) > 1 but this is not so after completion. Again this phenomenon lies at the very heart of Grothendieck-Teichmüller theory.

The (semi)local structure of C * and C P is not so mysterious. It is described in the following Proposition 2.9. Let v ∈ V (S) be a vertex of C * (S) and C P (S), with d(S) = k ≥ 0. Then v lies at the intersection of exactly k maximal copies of G (resp. F ) in C * (S) (resp. C P (S)). For any two copies G i ,

G j (i = j) one has G i ∩ G j = {v} ⊂ C * (S) and two vertices w i ∈ G i , w j ∈ G j with w i = v, w j = v are not joined by an edge in C * (S).
As for F , for any two copies F i and F j (i = j) we have F i ∩ F j = {v} ∈ C P (S) and for any w i ∈ F i such that v and w i are connected by an edge, the vertices w i and w j are not connected by a finite chain in C P (S).

Proof. Let v be given as a pants decomposition v = (α 1 , . . . , α k ). The main point here is that any triangle (complete graph on three vertices) of C * or C P is obtained by varying one of the α i 's keeping all the other curves α j fixed. This in turn depends only on the already mentioned (and obvious) fact that two curves on a surface of dimension 1 always intersect. So we get k copies of G inside C * which are indexed by the curves appearing in v. The rest of the statement and the transposition to C P is easily verified.

Note that this shows that d(S) can be read off (graph theoretically) from C * or C P . In fact it can be detected locally around any vertex v. To this end one can look for a star at v, namely a family (w i ) i∈I of vertices of C * (S) such that each w i is connected to v by an edge and no two distinct w i 's are connected. Then d(S) is the maximal possible number of such vertices i.e. the maximal cardinal of the index set I. Passing to C P (S), if w i , w j ∈ F i ⊂ C P (S), then there is a finite chain connecting w i and w j in the link of v. Together with the last assertion of the statement, this shows that there are exactly k = d(S) copies of F around v.

We now would like to reconstruct C(S) from C * (S), hence also from C P (S) by Lemma 2.7. One way to do this is to set up a correspondence between the subgraphs of C * (S) which are graph theoretically isomorphic to some C * (S σ ) (σ ∈ C(S)) and the subsurfaces of S. This correspondence, to be later adapted to the complete setting, is interesting even in this relatively simple discrete case. A precise wording goes as follows:

Theorem 2.10. Let C ⊂ C * (S) be a subgraph which is (abstractly) isomorphic to C * (Σ) for a certain surface Σ and is maximal with this property. Then there exists a unique σ ∈ C(S) such that C = C * (S σ ).

The proof is deferred to the next subsection. Here we list some fairly straightforward and important consequences. First one has: Corollary 2.11. C(S) can be (graph theoretically) reconstructed from C * (S).

Proof. Starting from C * (S) one builds a complex by considering subgraphs C as in the statement of the theorem, with the inclusion map as boundary operator. The result ensures that this simplicial complex is isomorphic to the curve complex C(S).

One then immediately gets:

Corollary 2.12. Aut(C * (S)) = Aut(C(S)).

Taking Lemma 2.8 into account, this shows that there is a natural injective map:

Aut(C P (S)) → Aut(C(S)),
from which by Theorem 2.1 we get the rigidity of the pants graph (a fortiori the pants complex) as Theorem 2.13. Let S be a hyperbolic surface of type (g, n) with d(S) > 1. Then the natural map θ P : Inn(M od(S)) → Aut(C P (S)) is an isomorphism.

For the fact that here type (1, 2) is no exception, see the last page of [START_REF] Margalit | Automorphisms of the pants complex[END_REF], of which we thus reproved the main result. We will see below (in §7) how this rigidity result (Theorem 2.13) does survive (procongruence) completion, in sharp and interesting constrast with item i) of Theorem 2.1) 2.3. Proof of Theorem 2.10. Let us start with some remarks and reductions. First we note that the word "maximal" is indeed necessary. For instance there are proper subgraphs of F (resp. G) which are isomorphic to F (resp. G). Second, implicit in the statement is the fact that any C * (S σ ) ⊂ C * (S) does indeed answer the problem, namely it is maximal in its isomorphy class. Assume on the contrary that we have a nested sequence C * (S σ ) ⊂ C ⊂ C * (S) where d(S σ ) = k, C is isomorphic to C * (S σ ) and the first inclusion is strict. Since C is connected, we can find a vertex w ∈ C \ C * (S σ ) which is connected by an edge to a vertex v ∈ C * (S σ ). Since S σ has dimension k, we can find k vertices w i ∈ C * (S σ ) as in the proof of Lemma 2.9 (with respect to v). But w ∈ C is connected to v and it is easy to check that it is not connected to any of the w i . In other words we have actually found k + 1 vertices which are connected to v and no two of which are connected, which contradicts the fact that C is isomorphic to C * (S σ ).

Having justified the statement, we can turn to the proof of Theorem 2.10, noticing first that uniqueness is clear: obviously C * (S σ ) coincides with C * (S τ ) (σ, τ ∈ C(S)) if and only if σ = τ ; this is also a very particular case of Lemma 2.6. From Lemma 2.9 we can now define

d(C) = d(Σ), which determines |σ| (assuming the existence of σ) since d(S σ ) = d(C) = d(S) -|σ|. Next the result is true if d(Σ) = 0 because then C * (Σ)
is just a point and so is C. Hence it does correspond to a vertex of C * (S), in other words to an actual pants decomposition of S. We will prove the result by induction on k = d(Σ) but it is useful and enlightening to prove the case k = 1 directly. This is easy and essentially well-known in a different context. Much as in Lemma 2.9 the point is that any triangle inside C * (S) (or C P (S)) determines a unique subsurface Σ with d(Σ) = 1. This sets up a one-to-one correspondence between subsurfaces of S of dimension 1 and maximal complete subgraphs of C * (S). Now let k > 1, assume the result has been proved for d(C) < k and consider a graph C ⊂ C * (S) as in the statement, with d(C) = k. We fix an isomorphism C ∼ -→ C * (Σ). Changing notation slightly for convenience, we are looking for a subsurface T ⊂ S, defined by a cell of C(S) and such that C = C * (T ). Note that it may happen that the surfaces Σ and T (assuming the existence of the latter) are not of the same type because of the well-known exceptional low-dimensional isomorphisms between complexes of curves. One will have C * (Σ) C * (T ) and indeed, as a consequence of the result itself, C(Σ) C(T ), so for instance Σ could be of type (0, 6) and T of type (2, 0).

We may now consider subsurfaces of Σ and transfer the information to C ⊂ C(S). Namely for any σ ∈ C * (Σ), we denote by C σ ⊂ C the subgraph corresponding to C * (Σ σ ) under the fixed isomorphism C C * (Σ). Actually, forgetting about this isomorphism, we just write

C σ = C * (Σ σ ) ⊂ C ⊂ C * (S)
. By the induction hypothesis, for any σ ∈ C(Σ), σ = ∅, there corresponds to C σ a unique subsurface S (σ) ∈ S. Beware of the fact that σ now runs over the cells of C(Σ), not of C(S), and this is the reason of the added brackets. In these terms we are trying to extend this correspondence to σ = ∅, i.e. find T = S (∅) .

In order to show the existence of T it is actually enough to show that there exists a k-dimensional subsurface of S, call it precisely T , such that any S (σ) with σ ∈ C(Σ) not empty is contained in T . Indeed, the corresponding C σ 's form a covering of C. So assuming the existence of such a subsurface T , we find that C ⊂ C * (T ); these two subgraphs being isomorphic and C being maximal by assumption, they coincide. In order to prove the existence of T , we can now restrict attention to the largest possible S (σ) 's, i.e. to the case |σ| = 1, which simply means that σ consists of a single loop.

We are thus reduced to showing that there exists a k-dimensional subsurface T ⊂ S such that, for any loop α on Σ, S (α) is contained in T . Now C(Σ) is connected because k > 1 and this can be used as follows. If α and β are two non intersecting curves on Σ, Σ α and Σ β are two subsurfaces of Σ of dimension k -1 intersecting along the subsurface Σ α∪β of dimension k -2, where α ∪ β is considered as a simplex of C(Σ). Informally speaking for the time being, the union S (α) ∪ S (β) has dimension k and this is the natural candidate for T . In other words the latter, if it exists, is determined by any two non intersecting loops of Σ. Returning to the formal proof, let γ and δ be two arbitrary loops on Σ. There exists a path in the 1-skeleton of C(Σ) connecting γ to δ. It is given by a finite sequence γ, α 1 , . . . , α n , δ of loops such that α 1 does not intersect γ, α n does not intersect δ and for 1 < i < n, α i does not intersect α i-1 and α i+1 . Using the existence of such a chain, we are reduced to the following situation. Let α, β, and γ be three loops on Σ such that α ∩ β = β ∩ γ = ∅; there remains again to show that S (α) , S (β) and S (γ) are contained in a common k-dimensional subsurface T , and this will complete the proof of the result.

We can write S (α) = S ρ , S The corresponding surfaces S ρ∩σ and S σ∩τ are both subsurfaces of S of dimension k. There remains only to show that they coincide: S ρ∩σ = S σ∩τ (= T ). We argue much as above, when proving that a subcomplex of type C * (S σ ) ⊂ C * (S) is maximal in its isomorphy class. The complexes C ρ∩σ and C σ∩τ are two subcomplexes of dimension k inside C which is also of dimension k, and they are maximal such complexes, being attached to subsurfaces of S. This forces them to coincide -and in fact coincide with the whole of C. More formally, assume the contrary, that is S ρ∩σ and S σ∩τ are distinct. Then, breaking the symmetry for a moment and relabeling if necessary, we can choose as above two vertices v ∈ C ρ∩σ and w ∈ C σ∩τ \ C ρ∩σ which are connected by an edge. Then again pick a maximal family (w i ) of k vertices in C ρ∩σ which are connected to v and are not mutually connected. Adding in the vertex w we get a family of k + 1 vertices with the same properties, which contradicts the fact that d(C) = k and completes the proof.

Profinite complexes and the isomorphism theorem

In this section we introduce and study profinite completions of the simplicial complexes which have appeared above. We focus on the procongruence completion because crucial results are not available to-date for the full profinite completions, as will become clear below. General foundations pertaining to completions of "spaces", possibly equipped with group actions, are now available in a profinite context, thanks in particular to the work of G.Quick who has put these objects in the classical framework of model categories (see [START_REF] Quick | Group action on profinite spaces[END_REF][START_REF] Quick | Some remarks on profinite completion of spaces[END_REF] and references therein). However in our much more specific context we can and do rely on the more direct constructions of the first author (see [START_REF] Boggi | Profinite Teichmüller theory[END_REF][START_REF] Boggi | On the procongruence completion of the Teichmüller modular groups[END_REF]). We then state and prove the crucial isomorphism result which very roughly speaking provides a bridge between group theoretic and complex or graph theoretic statements. We claim little novelty as to the framework and statements in this section, which are essentially borrowed from [START_REF] Boggi | On the procongruence completion of the Teichmüller modular groups[END_REF]. However some proofs in that paper (which itself uses [START_REF] Boggi | Galois coverings of moduli spaces of curves and loci of curves with symmetry[END_REF] in a crucial way) are not so easy to decipher and it thus seemed useful to provide at times alternative proofs or at least sketches thereof, using a more concrete, if somewhat ad hoc approach. We have also added a short "guide for the perplexed" ( §3.3) aiming at summarizing some of the main points of the theory, delineating a roadmap and pointing at a few serious bumps along the road.

3.1. Completions etc. Profinite complexes of curves were introduced in [START_REF] Boggi | Profinite Teichmüller theory[END_REF] ; the necessary constructions (and caveats) are summarized in [START_REF] Boggi | On the procongruence completion of the Teichmüller modular groups[END_REF], §3 to which we refer, especially concerning the congruence completions on which we focus hereafter. Minimal inputs appear in the Appendix below ( § §A. [START_REF] Fathi | Travaux de Thurston sur les surfaces (seconde édition)[END_REF][START_REF] Gibney | Towards the ample cone of Mg[END_REF]. Starting as usual from a (connected) hyperbolic surface of finite type S and the attending Teichmüller group Γ = Γ(S), one constructs in particular its (full) profinite completion Γ as well as its (pro)congruence completion Γ (see §A.10). One then proceeds to show that the cofinite subgroups Γ (m) ⊂ Γ (m > 2) pertaining to the abelian levels M (m) (see again §A.10 or [START_REF] Boggi | On the procongruence completion of the Teichmüller modular groups[END_REF] for much more) operate without inversion on the (discrete) curve complex C(S). This implies that this Γ-simplicial complex C(S) can be considered as a Γ (m) -simplicial set for any m > 2 (after numbering the vertices). Now by restricting to the congruence levels which dominate some such abelian level (that is the inverse system of congruence subgroups Γ λ with Γ λ ⊂ Γ (m) ⊂ Γ for some m > 2) we define the congruence completion Č(S) which we can view as a Γ-simplicial profinite set, that is a simplicial object in the category of profinite sets, which moreover is equipped with an action of the congruence completion Γ. We refer again to [START_REF] Boggi | Profinite Teichmüller theory[END_REF][START_REF] Boggi | On the procongruence completion of the Teichmüller modular groups[END_REF] for the necessary precisions. Roughly speaking this makes sense of the definition of the congruence completion as a pro-simplicial set defined by Č(S)

• = lim ←- λ∈Λ C(S) • /Γ λ
where Γ λ runs over the congruence subgroups of Γ, indexed by the (countable) set Λ. We denote the finite quotients by C λ (S) = C(S) • /Γ λ . Note that one may and it is sometimes useful to restrict consideration to the normal or even characteristic subgroups Γ λ since both types define cofinal inverse subsystems (because Γ is finitely generated). Note also that these completions are plainly defined "asymptotically", that is one can omit any subsequence of "large" subgroups. This is why for instance we may restrict to congruence subgroups which are contained in some subgroup Γ (m) (m > 2). So we regard Č(S) • as a simplicial object in the category of profinite sets, although below bullets are often omitted, while keeping in mind that we are indeed dealing with simplicial objects. There is a canonical inclusion C(S) → Č(S) ( [START_REF] Boggi | On the procongruence completion of the Teichmüller modular groups[END_REF], Prop. 3.3) with dense image and a natural continuous action of Γ on Č(S).

In a similar fashion and for the same reasons we can define ČP (S) as the inverse limit

ČP (S) • = lim ←- λ∈Λ C P (S) • /Γ λ
and regard it again as a simplicial object in the category of profinite sets. It is in fact a prograph; the finite quotients are denoted C λ P (S) = C(S) • /Γ λ . There is again a canonical inclusion C P (S) → ČP (S) with dense image, which is equivariant for the Γ-action (resp. Γ-action) on C P (S) (resp. ČP (S)) and the inclusion Γ(S) → Γ(S). Finally, as in the discrete case, there is a one-to-one correspondence between the vertices of ČP (S) and the simplices of Č(S) of maximal dimension (= d(S) -1). A deep additional information is contained in the edges of ČP (S).

We now concentrate on alternative, more geometric and manageable descriptions of the congruence curves complex Č = Č(S). More precisely we will shortly define the simplicial profinite complexes ČL = ČL (S) and ČG = ČG (S), denoted respectively L(π) and L (π) in [START_REF] Boggi | On the procongruence completion of the Teichmüller modular groups[END_REF] (π = πtop 1 (S), the topological fundamental group of the surface S) to which we refer for more detail. An important result, stated and proved in the next subsection asserts that Č(S), ČL (S) and ČG (S) are isomorphic, so that we are indeed describing the same object from several standpoints. Typically, these three objects can be defined in the full profinite setting but the fact that they are isomorphic is not known.

In order to define ČL (S), where L stands for "loops" we first define its set of vertices L(S) = ČL (S) 0 , the set of unoriented proloops. Recall that in the discrete setting L(S) = C L (S) 0 denotes the set of unoriented simple loops up to isotopy which moreover are not peripheral, that is do not bound a disc on S with a single puncture. We are looking for a completion which is a priori simpler and more manageable than the one afforded by Ĉ(S) in that it will involve only the fundamental group π = π top 1 (S) and its completion, instead of the much more involved Γ = π top 1 (M(S)), the topological or orbifold fundamental group of the moduli space of curves.

We proceed as follows (see again [START_REF] Boggi | On the procongruence completion of the Teichmüller modular groups[END_REF], §3). For a set X, let P(X) denote the set of unordered pairs of elements of X and for G a group, let G/ ∼ denote the set of conjugacy classes in G. Now given γ ∈ π, denote by γ ± the equivalence class of the pair (γ, γ -1 ) in P(π) and by [γ ± ] the equivalence class of γ ± in P(π/ ∼). Note that the latter has a natural structure of profinite set. The point is that there is a natural embedding ι : L → P(π/ ∼). Indeed, given a loop ∈ L, it can be represented by an element γ = γ( ) ∈ π and we define ι( ) = [γ ± ], which is plainly independent of the choice of the representative γ of . Finally we define the set L = L(S) of proloops on S as the closure of the image ι(L) inside P(π/ ∼), where we are using the nontrivial fact from combinatorial group theory (conjugacy separability for the group π) that the natural map P(π/ ∼) → P(π/ ∼) is injective.

It is then easy to define, in much the same way, the simplicial complex C L (S) (with L = C L (S) 0 ) and its completion ČL (S) (with L = ČL (S) 0 )). For X a set and k ≥ 1, define P k (X) to be the set of unordered subsets of P(X) with k elements (P = P 1 ). Then we get a natural embedding ι k : C(S) k → P k+1 (π/ ∼) (ι = ι 0 ) of the k-simplices of the discrete curve complex into the unordered sets of k + 1 conjugacy classes of the group π modulo inversion. There remains only to define ČL (S) k as the closure of the image and to organize the collection of profinite sets ( ČL (S) k ) (0 ≤ k ≤ d(S) -1) into the simplicial complex C L (S) • , using the usual face and degeneracy operators (deleting and adding elements).

The last avatar ČG (S) of the congruence complex is actually easier to define. It is enough to define its sets of vertices ČG (S) 0 and then proceed as above. Return to L(S); mapping a simple loop to the cyclic subgroup of Γ generated by the corresponding twist, we get a natural embedding L → G(π)/ ∼ where the right-hand side denotes the set of cyclic subgroups of π modulo conjugacy. Again we have a further natural injective map G(π/ ∼) → G(π)/ ∼ and we denote by Ĝ(S) the closure of the image of L in G(π)/ ∼ via the composite embedding. Equivalently we may consider the image G(S) of L in G(π)/ ∼ and then take its closure Ĝ(S in G(π)/ ∼, corresponding to certain procyclic subgroups of π, still up to conjugacy. Starting from Ĝ(S) = ĈG (S) 0 we then build up the prosimplicial complex ČG (S) the same way we built ČL (S) out of L(S).

The next subsection will be essentially devoted to showing that these three avatars of Č(S) (including Č(S) itself) are isomorphic. Here in closing we add a few simple but extremely useful remarks about this type of relatively new objects. First of all one should keep in mind that we are dealing with compact (totally disconnected) spaces. This means in particular that there is no "going to infinity". As a first extremely crude approximation Ĉ(S) or Č(S) differ as much from C(S) as the ring Z p of the p-adic integrs differs from Z. Note for instance that Thurston's theory precisely starts from considerations connected with geometric intersection numbers, twists and ways of going to infinity, whether on Teichmüller space T (S) or on the curves complex C(S). Nothing of the kind is available -nor even relevant -here. For much more on a dynamical viewpoint on these objects we refer to [START_REF] Lochak | Results and conjectures in profinite Teichmüller theory[END_REF], §8.

Next we sketch a line of arguments which we will meet below more than once. Let X(= X • ) denote a discrete G-simplicial complex with G a finitely generated group. Assume the number of G-orbits in X is finite. Let G be some completion of G and assume we have constructed a completion X of X which is a G -prosimplicial complex. In particular there is a natural morphism ι : X → X with dense image and X enjoys the universal property that any morphism φ : X → Z from X to a G -prosimplicial complex Z factors through a unique φ : X → Z i.e. φ = φ • ι. Moreover ι is equivariant for the G-action on X and G -action on X . Note that all the morphisms we consider are continuous for the natural topologies on their respective source and target.

In the situation above, there is at first a seemingly simple description of X which goes as follows. Pick k ≥ 0 and let X k denote the k-skeleton of X ; by assumption one can decompose X k into disjoint G-orbits enumerated by the finite set E k :

X k = σ∈E k G • σ,
where the k-simplices σ are representatives in the orbits. Under these circumstances one can decompose the k-skeleton X k of X as

X k = σ∈E k G • ι(σ).
In other words it is covered by the G -orbits of the images of the same simplices. Note that these orbits now may not be disjoint. In all the cases we will encounter X is residually finite, that is ι is injective, and we omit it from the notation. So X is made of (not necessarily disjoint) G -orbits and there are finitely many in every dimension. The one line proof of the above is both simple and instructive. Consider the right-hand side of the equality above: it is compact because so is G and E k is finite; it is dense because it contains ι(X). So it coincides with X k .

Finally let X and Y be as above and for simplicity assume they are both residually finite so that we identify X (resp. Y ) with its image in X (resp. Y ). Let f : X → Y be a simplicial morphism. It naturally determines a morphism f : X → Y by the universality property of the completion. Moreover, if f is onto, so is f . The proof is again one line : the image f (X ) contains f (X) = f (X) = Y ⊂ Y which is dense ; f (X ) being dense and compact (as the continuous image of a compact) in Y , it coincides with it.

3.2. The isomorphism theorem. Let us return to S and the three attending versions of the congruence complex, namely Č(S), ČL (S) and ČG (S). By the universality of the Γ-completion we have a sequence of (simplicial) maps : Č(S) → ČL (S) → ČG (S).

We can now apply (twice) the reasoning immediately above (end of §3.1) and conclude that both maps are surjective. Their injectivity constitutes one of the main statements in [START_REF] Boggi | On the procongruence completion of the Teichmüller modular groups[END_REF] :

Theorem 3.1 ([4], Theorem. 4.2)
. The natural maps Č(S) → ČL (S) and ČL (S) → ČG (S) are Γ-equivariant isomorphisms of prosimplicial sets.

Sketch of proof. We will present a partial proof of this important result (using ideas from [START_REF] Boggi | On the procongruence completion of the Teichmüller modular groups[END_REF]), breaking it into three propositions. First it is clearly enough to show that the composition of the two maps is injective and one can actually restrict to showing that the map on the vertices, namely

Φ : Č(S) 0 → ČG (S) 0 ,
is injective, hence a bijection since it is known to be surjective. Recall that on the left Č(S)

0 = Ľ = lim ←- λ∈Λ L/Γ λ
where λ ∈ Λ runs over the congruence subgroups of Γ. The right-hand side is given as the closure of the set of cyclic subgroups of π corresponding to elements of L(S) inside G(π)/ ∼, the set of procyclic subgroups of π (π = π top 1 (S)) modulo conjugacy. Both sides are naturally equipped with a Γ-action and the map Φ is equivariant and onto. The only moot point is injectivity, whose validity is equivalent to that of the statement of the theorem. We used the symbol Ľ(S) because L(S) has already been used for the set ČL (S) 0 of vertices of ČL (S); a posteriori the theorem will confirm that Ľ(S) = L(S).

Our first assertion reads:

Proposition 3.2. The map Φ induces a bijection between the respective Γ-orbits of Č(S) 0 and ČG (S) 0 .

In fact these Γ-orbit have nothing mysterious. Indeed recall how curves and (Dehn) twists are related with the Γ-action in the discrete case. If α ∈ L is a loop (i.e. an isotopy class of simple closed curves), τ α the associated twist (we assume that the surface S has been given an orientation once and for all) and g ∈ Γ, then we have the familiar and elementary formula

τ g•α = gτ α g -1 .
Anticipating (a lot) we remark that the Grothendieck-Teichmüller action can and will be seen essentially as a generalization of this formula to 'procurves' and 'protwists'. For the moment we recall that this provides a description of the Γ-orbits of the discrete complex C(S) = C L (S) = C G (S) (with obvious definitions). Two loops α and β lie in the same Γ-orbit if and only if the topological types of the two slit surfaces S α = S \ α and S β = S \ β coincide. This is also the necessary and sufficient condition for the two associated twists τ α and τ β over α and β to be Γ-conjugate. The topological type of a twist τ γ along a curve γ is defined as the type of S γ , the surface S slit along γ, which we also refer to as the type of the curve γ itself. Now any Γ-orbit in Č contains a discrete representative, i.e. a curve in L (see the end of §3.1). So the Γ-orbits of Č are enumerated, with possible redundancies, by the finitely many topological types of the slit surfaces S α (α ∈ L), which also enumerate the irreducible components of the divisor at infinity of the stable compactification of M(S). The same is true of the Γ-orbits of ČG , for the same reason. Since Φ is onto, this shows that Proposition 3.2 is a consequence of the following:

Proposition 3.3. Given twists τ α , τ β ∈ Γ ⊂ Γ (with α, β ∈ L) two nontrivial powers τ k α , τ β (k, ∈ Ẑ \ {0}
) are conjugate in Γ if and only if k = and τ α and τ β (equivalently α and β) have the same topological type.

Note that this will show that the topological type of a "protwist", or actually a power thereof is welldefined as the type of any discrete twist lying in the same Γ-orbit, a protwist being nothing but a Γ-conjugate of some bona fide discrete twist. We will henceforth often skip the prefix "pro" ("protwists", "procurves", etc.) when it should not lead to confusion. We also remark that it has long been known that the congruence levels separate the powers of a twist (or protwist for that matter). That is, given a twist τ , the natural map Ẑ → Γ which sends a ∈ Ẑ to τ a is injective. In other words the procyclic group τ α generated by a twist τ α is contained in Γ.

Granted Proposition 3.3 (see below for its proof) there remains to show what we state as:

Proposition 3.4. For every α ∈ L = C(S) 0 ⊂ Č(S) 0 the Γ-stabilizer Γα ⊂ Γ of α as an element of Č(S) 0 coincides with the stabilizer of its image Φ(α) ∈ ČG (S) 0 .

Here again one can reduce -as we did -the question to the stabilizer of a discrete curve by first acting with Γ. Moreover, by [START_REF] Boggi | Profinite Teichmüller theory[END_REF], Proposition 6.5, the Γ-stabilizer of α is the closure in Γ of the stabilizer Γ α ⊂ Γ of α ∈ C(S) 0 , that is viewed as an element of the discrete complex C(S). Finally the discrete stabilizer Γ α affords an elementary geometric description.

We have now reduced the proof of Theorem 3.1 to those of Propositions 3.3 and 3.4. We will present the first in detail, partly for its own sake, partly in order to illustrate certain techniques in a concrete, if somewhat ad hoc way. By contrast, we will essentially rely on [START_REF] Boggi | On the procongruence completion of the Teichmüller modular groups[END_REF] for the proof of Proposition 3.4.

Proof of Proposition 3.3. First let us clarify the (natural) definition of a profinite power. If -say -g ∈ Γ and k ∈ Ẑ, then g k ∈ Γ is defined explicitly as an inverse system. For a level λ ∈ Λ, let a λ denote the order of the finite group Γ/Γ λ . Then the λ-component of g k reads g λ k λ where g λ ∈ Γ/Γ λ is the λ-component of g and k λ ∈ Z/a λ is the a λ -component of k (of course this definition is valid for any completion of any group).

Consider again τ α and τ β , where α, β ∈ L(S). We need only prove the only if part of the statement: given two profinite powers τ k α and τ β (k, ∈ Ẑ \ {0}) there should exist a finite congruence quotient of Γ in which their images are not conjugate, if either α and β do not share a common type, or k and are different.

For α ∈ L(S), let T α denote the action in homology of τ α . It is well-known that for any loop γ ∈ L(S) on the surface we have

T α [γ] = [γ] + [γ], [α] [α]
where [γ] denotes the homology class of the curve γ and , is the symplectic intersection form on S. Therefore T α is either trivial, when the curve is separating (i.e. when [α] = 0), or it can be represented by a nontrivial elementary matrix with one unit nonzero entry outside of the diagonal, when the curve α is nonseparating. Therefore the conjugacy classes of (powers of) twists along two curves, at least one of which is non separating, can be distinguished in any nontrivial congruence quotient of the integral symplectic group. We are thus reduced to the case where both α and β are separating, which we assume from now on. Let f : S → S be a characteristic (finite unramified) cover associated to a finite index characteristic subgroup K = π 1 ( S) ⊂ π = π 1 (S), which can be identified with the image f * (π 1 ( S)) by the map f * induced at the level of fundamental groups. It is Galois with Galois group G K = π/K. We would like to compute the action in homology of the lift of a twist to such a cover. If φ ∈ Aut(π) is an automorphism of π, its restriction to K determines an automorphism φ ∈ Aut(π)) (π = π 1 ( S) = K), via the requirement of equivariance

f * • φ = φ • f * .
This also defines a way of lifting mapping classes ϕ ∈ Out + (π) = Γ = Γ(S) (where the superscript + indicates the preservation of orientation) to φ ∈ Out + (π) = Γ = Γ( S); the lift is well-defined up to the action of the Galois group G K . Since mapping classes are determined by their action on the simple closed curves we derive that φ is determined -again up to multiplication by an element of G K -by the equivariance on these, that is the property that

f ( φ(γ)) = ϕ(f (γ))
(up to homotopy) for every γ ∈ L(S) and γ ∈ L( S) with γ = f * (γ).

We say that a loop α ∈ L( S) on S is a lift of α ∈ L(S) if it is a connected component of its preimage f -1 (α); any two lifts of any two equivalent curves (i.e. curves with the same type) are equivalent. We denote by α n the curve obtained by traveling n times around α. If α is a lift of α, the restriction of f to α defines a finite covering of α of degree -say -m(α), which is independent of the choice of the lift, indeed only depends on the type (i.e. equivalence class) of α. In fact m(α) coincides with the order in G K = π/K of any element of π represented by the curve α, which can be seen as follows. Let d(α) denote this order; it is well-defined since the various elements representing α belong to a single conjugacy class of G K . Then (the class of) α d(α) belongs to K = f * (π 1 ( S)) and hence it can be lifted to a closed curve α. Moreover α cannot be a power of some other curve β (α = βn , n > 1) because if so the restriction of f to β would cover α with degree d(α)/n < d(α). Hence α ∈ L( S) and m(α) = d(α).

We wish to describe a lift τα of τ α to Γ, or at least a suitable power of it. So let γ ∈ L(S), γ ∈ L( S) a lift of γ. By the above

f (τ α (γ)) = (τ α (γ)) d(γ) .
On the other hand it also holds that

f (τ α(γ)) = (τ d(α) α (γ)) d(γ)
from which we conclude that τ d(α) α = τ α.

We can now compute the action of τ d(α) α on the homology of S, which we denote T d(α) α

. Indeed the action of any power is determined by

T kd(α) α [γ] = T k α [γ] = [γ] + k [γ], [α] [α]
with k an integer, γ ∈ L( S) and the angle brackets denote the symplectic pairing on S (here and below pairings will implicitly relate to the relevant surface). From the above we find in particular that

T kd(α) α [γ] = [γ] + k [γ], [ α] [α].
Raising this identity to the power d(β)) we find that

T kd(α)d(β) α [γ] = [γ] + kd(β) [γ], [α] [α]; swapping (α, k) and (β, ), this delivers T d(α)d(β) β [γ] = [γ] + d(α) [γ], [ β] [ β].
We now choose a basis of the integral homology group H 1 ( S) = H 1 ( S, Z) as follows. First consider the curves α and β along with their images by the deck transformation group G K ; we then further adjoin simple closed curves to this set until we reach a maximal set of pairwise disjoint curves which is invariant under the action of G K , so that no two curves are pairwise homotopic (and none is null homotopic).

Assume that the covering S is such that the lifts α and β of both original curves α and β are nonseparating on S. The validity of this crucial assumption will be discussed below. Granted this for the moment and using the basis of H 1 ( S) described above, the matrices corresponding to

T kd(α)d(β) α , resp. T d(α)d(β) β read 1 B α 0 1 resp. 1 B β 0 1 where B α (resp. B β ) is a diagonal matrix having d = |G K | nonzero entries equal to kd(α) (resp. d(β)).
Note that G K acts via permutations on the basis of H 1 ( S). Assume first that k = but α and β are not conjugate, and consider the principal congruence quotients of the integral symplectic group Aut(H 1 ( S), , ) of the form Aut(H 1 ( S, Z/mZ), , ) with m an integer.

We will show below that there exists an m such that the image of

T kd(α)d(β) α is not conjugate to any matrix in the G K -orbit of T kd(α)d(β) β
, which will imply that the images of

τ kd(α)d(β) α = τ k α d(α)d(β) and τ kd(α)d(β) β = τ k β d(α)d(β) are not conjugate in the image of Γ in Aut(H 1 ( S; Z/mZ), , )/G K . Finally it is
known that the latter is a congruence quotient of Γ, which will complete the proof of the proposition in that case. The other case, when k = but α and β are conjugate, is easy (one can assume that α = β).

In order to find a cover S → S as described above, it is enough to find a characteristic subgroup K such that d(α) and d(β) are mutually prime. Indeed, picking then m = d(β) above, the image of T kd(α)d(β) β will be the identity modulo m along with all its G K -conjugates, whereas the image of T km(α)m(β) α will be a nontrivial unipotent, provided the curve α is non separating on S. Summarizing the above, in order to complete the proof of the proposition there remains to find a characteristic cover S → S such that the lifts of α (and of β as well, in order to preserve symmetry) are nonseparating, whereas d(α) and d(β), that is the respective orders of the lifts of α and β in the group of the cover, are coprime; here recall that the lifts of a given loop, i.e. the connected components of its preimage, are conjugate in the group of the cover.

The two requirements above are essentially independent. First note that for any cover, the lifts of the nonseparating loops are nonseparating. Next it turns out to be easy to exhibit covers of S in which the lifts of all the separating loops, hence all the loops, are nonseparating (see below). Furthermore the lifts of the separating loops are simple : d(α) = 1 for any separating α ∈ L(S). (Note that here we are actually dealing with conjugacy classes of loops since the elements of L(S) are not attached to a base point, but that does not affect the argument.) Then any further cover has the property that all the lifts are nonseparating and there remains to manufacture such a cover with d(α) and d(β) coprime. We can thus break the remaining part of the proof of the proposition into two lemmas, the first of which reads: Lemma 3.5. For any integer m ≥ 1, consider the cover S (m) corresponding to the invariant subgroup π (m) which is the kernel of the natural surjection

p (m) : π = π 1 (S) → H 1 (S, Z/m).
Then the lifts of all the loops on S to S (m) are non separating and simple.

Proof. Let α ∈ L(S) be a loop on S. If α is non separating there is nothing to prove. If it is, then its image [α] in homology is trivial, and so in particular is its reduction in H 1 (S, Z/m). In other words π (m) contains all the separating loops. So we find that d(α) = 1 which, referring to the above, implies that the multiplicity (= d(α)) of any connected component of the preimage p -1 (m) (α) is also equal to 1. But this says that this preimage breaks into d m = |H 1 (S, Z/m)| non separating curves, whose union separates S (m) .

Here are some additional remarks. First the covers S (m) are precisely those which are used when defining the abelian levels M(S) (m) and the principal congruence subgroups Γ(S) (m) ⊂ Γ(S) (see e.g. [START_REF] Boggi | Galois coverings of moduli spaces of curves and loci of curves with symmetry[END_REF], §1). Then [START_REF] Boggi | Galois coverings of moduli spaces of curves and loci of curves with symmetry[END_REF] (whose proof is much trickier) asserts that all the covers associated to such subgroups (under some mild additional conditions) have the property that the inverse image of a loop does not contain separating loops. Indeed it states much more which we refrain from detailing here. This provides a much larger sample of covers and constitutes the basis for the essential "linearization" of the tower of congruence subgroups of Γ(S) (see §3.3 below). Finally and in a different vein, we note the tantalizing analogy between the breaking of the preimage of separating loops in a Galois cover and the completely split primes in a Galois field extension. We now turn to the second and concluding lemma namely: Lemma 3.6. Given α, β ∈ L(S) there exists a finite, unramified, Galois cover of S such that d(α) and d(β) are coprime.

π (m) = [π, π] • π m is a cofinite invariant subgroup of π and so is K (m) = [K, K] • K m for any cofinite invariant subgroup K ⊂ π. Lemma 3.10 in
Proof. Fix two coprime numbers and m. Consider the quotient group

Π = Π ( ,m) = π 1 (S)/ α = 1, β m = 1
This is the fundamental group of a 2-complex obtained by adding 2-cells along the relations. It splits as an amalgamated product Π = Π 1 * α Π 2 * β Π 3 where:

Π 1 = π 1 (S 1 )/ α = 1 , Π 2 = π 1 (S 2 )/ α = β m = 1 , Π 3 = π 1 (S 3 )/ β m = 1 .
Notice now that the Π j 's are fundamental groups of orbifolds, namely they are Fuchsian groups of nonzero genus. In particular they are conjugacy separable, hence they admit finite quotients in which α has order and β has order m. Pick such finite quotients Q j of Π j , so that Π surjects onto an amalgamated product

Q = Q 1 * α Q 2 * β Q 3 .
It is well-known that a graph of groups in which the vertex groups are finite, such as Q, is virtually free. Let now Q denote a finite quotient of Q such that ker(Q → Q) is free. Then the images of α and β have respective orders and m in Q, since the kernel is torsionfree.

Consider next a finite index characteristic subgroup K of π contained in ker(π → Q), for instance the intersections of its images by all the conjugacy automorphisms. Then G K surjects onto Q and in particular the orders of α and β in G K are divisors of and m, respectively. In particular, these are coprime integers.

This completes the proof of the lemma, hence also of Proposition 3.3.

As mentioned above we refer to [START_REF] Boggi | On the procongruence completion of the Teichmüller modular groups[END_REF] for the proof of Proposition 3.4, which will complete the proof of Theorem 3.1. In fact the core of the proof of Proposition 3.4, to be found at the very end of the proof of Theorem 4.2 in [START_REF] Boggi | On the procongruence completion of the Teichmüller modular groups[END_REF] (top of p.5200) consists in a direct application of the "linearization theorem" in [START_REF] Boggi | Galois coverings of moduli spaces of curves and loci of curves with symmetry[END_REF], to which we return in the next subsection. In essence it does not differ so much from the proof of Proposition 3.3 presented above, which is in line with the proof of the linearization theorem.

Thanks to the isomorphism theorem we will henceforth often refer to the (pro)congruence curve complex, without explicitly distinguishing between its three versions, namely Č(S), ČL (S) and ČG (S). As a last item in this paragraph we mention a fairly direct consequence of Proposition 3.3, namely: Proposition 3.7. The Γ(S)-orbits of the simplices of the procongruence complex Č(S) are in one-to-one correspondence with the Γ(S)-orbits of the simplices of the discrete complex C(S).

Proof. It is enough to show that if two discrete (k -1)-simplices α = {α 1 , . . . , α k } and β = {β 1 , . . . , β k }, as viewed in Č(S), sit in the same Γ-orbit, then they actually belong to the same Γ-orbit. Proposition 3.3 takes care of the case of loops (k = 1) and then one proceeds by induction. Assuming α and β are in the same Γ-orbit, Proposition 3.3 says there exists g ∈ Γ such that g(α 1 ) = β 1 . After twisting by g we may thus assume that α 1 = β 1 . Now by assumption there exists h ∈ Γ such that h(α) = β and h belongs to Γα1 , the stabilizer of the loop α 1 in Γ. By [START_REF] Boggi | On the procongruence completion of the Teichmüller modular groups[END_REF] (Theorem 4.5) this stabilizer is naturally isomorphic to an extension of Γ(S α1 ) by the procyclic group τ α1 generated by the twist along α 1 . Here S α1 denotes as usual the surface S slit along the loop α 1 and note that we are using the fact that we consider precisely the procongruence completion (see [START_REF] Boggi | Profinite Teichmüller theory[END_REF], Proposition 6.6). Multiplying out by a (profinite) power of the twist τ α1 , we are led to dealing with (k -2)-simplices on the surface S α1 , where the assertion holds true by induction, which proceeds either on the dimension of the simplices or on the modular dimension of the underlying surface S.

3.3.

Elucidation. Before moving forward it may be desirable, indeed necessary, to elucidate the actual content of the above isomorphim result and its significance. The point is roughly that objects which are more or less clearly equivalent (isomorphic) in the discrete case, are definitely not obviously so after completion. Sometimes the equivalence requires a difficult proof and sometimes it simply does not hold true. So let us first briefly review the various objects connected with isotopy classes of simple closed curves (a.k.a. loops) on a connected oriented hyperbolic surface S. We will essentially confine ourselves to the case of a single loop, higher simplices are determined by their vertices.

Let us first summarize and review four constructions, starting from an oriented loop γ on S, where we may consider that γ ∈ π = π 1 (S). Since π is constructed picking out a basepoint P ∈ S this means that we choose a loop through P in the free isotopy class of γ. Let γ ∈ L(S) = C(S) 0 denote γ after forgetting the orientation.

Working again with the fundamental group π, specifying γ is equivalent to specifying a pair γ ± of two oriented loops with opposite orientations. Passing to conjugacy classes in order to free the construction from the choice of a basepoint, we find that γ ∈ L leads to an unordered pair [γ ± ] of elements of π/ ∼ which is now an element C L (S) 0 .

That was so to speak on the graph theoretic side. Now from a group theoretic viewpoint, γ defines the cyclic subgroup γ ⊂ π it generates inside π ( γ and γ -1 define the same subgroup). Considering the subgroup γ up to conjugacy in π leads to the definition of γ as an element of C G (S) 0 . Slightly more generally, given any integer k > 0, one can consider the finite index cyclic subgroup γ k ⊂ π. This will prove useful below.

Finally one can pass to the Teichmüller group Γ(S). Then γ defines the twist τ γ along it (using the orientation of S) and again the cyclic group τ γ ⊂ Γ(S) or its finite index subgroups τ k γ ⊂ Γ(S) (k > 0). So far so good in the discrete case. Part of the fundational work then consists in exploring what happens after completion. A main point is that one can complete either working directly with π, the fundamental group of the surface S, and thus its profinite completion π, or with Γ = Γ(S), the fundamental group of the moduli space M(S). It is clear a priori that these two forms of completions can be related only if one considers completions of Γ that are no finer than the congruence completion Γ, which records the covers of M(S) coming from covers of S. Recall that the congruence conjecture asserts that in fact Γ = Γ. So in some sense the problem, from this foundational standpoint, consists in setting up a dictionary between these two kinds of completions, and also, in a slightly different but closely related fashion, between the graph theoretic and the group theoretic information.

Concretely, what are then the main tools and results ? We will list one essential tool and two foundational results, globally referring to [START_REF] Boggi | On the procongruence completion of the Teichmüller modular groups[END_REF][START_REF] Boggi | Galois coverings of moduli spaces of curves and loci of curves with symmetry[END_REF]. Let us give these threee statements names as it can help further reference as well as pointing to the core of the matter. The tool leads to a kind of linearization of the problem, replacing homotopy with homology. The first result is precisely the isomorphism theorem above (Theorem 3.1) ; the second one expresses a property we will refer to as twist separability. Let us now go into somewhat more detail.

The idea of "linearization" is fairly old and may be ascribed to E.Looijenga. It has actually been used in the proof of Proposition 3.3 above. A general expression of this principle is embodied by Corollary 7.8 in [START_REF] Boggi | On the procongruence completion of the Teichmüller modular groups[END_REF]. A proper statement is cumbersome and requires introducing a lot of notation, so let us content ourselves with the main idea, namely that given S as above, a loop γ ∈ L(S) is entirely determined by the projective set of the homology classes of its preimages on the (finite unramified) covers of S. Explicitly and with π = π 1 (S), let K ⊂ π an invariant finite index subgroup (normal would be enough but invariant is forced when working with mapping class groups), let G K = π/K denote the quotient group, p K : S K → S the ensuing Galois cover with group G K . For α ∈ L(T ) a loop on a surface T , let [α] ∈ H 1 (T, Z) denote the associated integral homology class. Then given γ ∈ L(S), we can consider the projective system ([p -1 K (γ)]) K of homology classes on S K , where K runs through the cofinite invariant subgroups of π (for K = π, S K = S and we omit the mention of p π = id). Roughly speaking, the theorem asserts that γ is entirely determined by the family of "linear" data ([p -1 K (γ)]) K . What are the obvious obstacles which arise when trying to identify a loop via its homology class? In fact, a loop α ∈ L(S) is trivial in homology, that is [α] = 0 ∈ H 1 (S), if and only if α is separating. More generally, given non intersecting loops α, β ∈ L(S), their homology classes coincide ([α] = [β]) for the appropriate orientations if and only if they form a cut pair, that is their union separates the surface (the first case can be seen as the case β = ∅). This is why it is important to detect a large sample of covers p K : S K → S such that for any loop on S, more generally any simplex σ ∈ C(S), the inverse image p -1 K (σ) does not contain separating curves nor cut pairs. This is provided by the important Lemma 3.10 in [START_REF] Boggi | Galois coverings of moduli spaces of curves and loci of curves with symmetry[END_REF] (see above, after the proof of Proposition 3.3).

Passing to the first main result, it was already mentioned that it is embodied by the isomorphism theorem (Theorem 3.1) above. Here we simply insist again that its main thrust lies in connecting, on the one hand completion via the Teichmüller group Γ(S) i.e. the fundamental group of the moduli space M(S), which is used when defining Č(S), on the other hand completion via the much simpler and more tractable fundamental group π = π 1 (S) of the surface S itself, which is used when defining both ČL (S) and ČG (S).

The second main result traces a fundamental link between (pro)curves and (pro)twists, that is between the graph theoretic and the group theoretic facets of the theory. This is Theorem 5.1 in [START_REF] Boggi | On the procongruence completion of the Teichmüller modular groups[END_REF], which can be stated more easily. Starting in the discrete setting we have (after orienting the surface S) a natural injective map d : L(S) → Γ(S) which to a loop γ ∈ L(S) assigns the corresponding twist τ γ . Given k ∈ Z \ {0} it can be generalized to

d k : γ → τ k γ (d = d 1 )
, still an injective map between the same source and target. As usual, upon completion the plot thickens and things become more interesting. From the injective map d and the natural embedding Γ → Γ we get a (still injective) map which we denote by the same name for simplicity d : L → Γ. By the universality of the progruence completion, this leads to a map d : Ľ(S) → Γ(S) which now may or may not be injective (this is precisely the moot point here) with, as above,

Ľ(S) = Č(S) 0 = lim ←- λ∈Λ L(S)/Γ λ .
Finally the isomorphism theorem ensures that Ľ(S) = L(S) = ČL (S) 0 , namely the set of (pro)curves on S. This can be generalized in the obvious way to dk : Ľ(S) → Γ(S) for any k ∈ Z \ {0} and indeed jazzed up to k ∈ Ẑ \ {0}, using the density of Z in Ẑ. Note from a topological viewpoint that in the complete case we are always considering continuous maps between compact spaces. We may now state the second fundamental result about twists separability we have been alluding to: In words : a (pro)curve can be detected via any (profinite) power of the associated twist. Indeed more is true, as can be gathered from the -difficult -proof of the above result (see [START_REF] Boggi | On the procongruence completion of the Teichmüller modular groups[END_REF], Remark 5.14). Let D k ⊂ Γ(S) denote the set of k-th powers of twists (k ∈ Z \ {0}) and let Ďk ⊂ Γ(S) denote its closure in Γ(S). Extend the definition of Ďk to k ∈ Ẑ \ {0}. Then one can show that the intersection Ďk ∩ Γ(S) is exactly D k if k ∈ Z and is empty if not. This leads to the following striking corollary of the above theorem, or rather of its proof, which will be substantially strengthened below (see in particular Proposition 4.3): Corollary 3.9. Let α, β ∈ L(S) be two (pro)curves, k, ∈ Ẑ \ {0} two nonzero (pro)integers, then the equality τ k α = τ β holds if and only if α = β and k = . We will next proceed to record some important consequences of these foundational results, before moving to the study of the automorphisms of our various simplicial complexes. Note again that these complexes are entirely determined by their 1-skeleta. So in some sense we are primarily interested in profinite graphs.

Remark 3.1. The above is in some ways reminiscent of anabelian geometry and the exploration of the structure of the Galois groups of fields, especially fields of functions (including in positive characteristic). On this topic we refer for instance to [START_REF] Saïdi | On the Hom-form of Grothendieck's birational conjecture in positive characteristic[END_REF] and in particular some basic phenomena summarized there in Proposition 1.5, as well as to the work of F.Pop (starting with [START_REF] Pop | Glimpses of Grothendieck's Anabelian Geometry[END_REF]).

From Graphs to Groups and back : Centralizers and Normalizers of twists

Fixing as usual a connected oriented hyperbolic surface S, there is a natural action of Γ(S) on the three isomorphic versions of the curve complex, namely Č(S), ČL (S) and ČG (S). This group also acts on powers of twists by conjugation. By Theorem 3.8, given a loop α ∈ L(S), the centralizer Z(τ α ) of τ α in Γ coincides with the stabilizer Γα of α for the action of Γ on Ľ = L. This still holds true for τ k α with α ∈ L and k ∈ Ẑ \ {0}. Now the stabilizer Γα admits a rather explicit description, and more generally so does Γσ , the stabilizer of a simplex σ ∈ Č(S) (equivalently ČL (S), ČG (S)). The structure is identical to the one occurring in the discrete case and this can be vindicated relatively easily; see [START_REF] Boggi | On the procongruence completion of the Teichmüller modular groups[END_REF], Theorem 4.5 and references there. It is important to insist at this point that we are definitely using the procongruence completion. The analogous description for the full profinite completion (as stated in [START_REF] Boggi | Profinite Teichmüller theory[END_REF]) remains unproved to-date.

Here we give a short and partial account of the descriptions of the centralizers and normalizers of twists as well as of the commutative subgroups of Γ(S) generated by finite sets of commuting twists. We refer globally to [START_REF] Boggi | On the procongruence completion of the Teichmüller modular groups[END_REF][START_REF] Boggi | A restricted Magnus property for profinite surface groups[END_REF] for detailed results and proofs. The second reference improves on the first, addressing in particular the case of multitwists, that is products of powers of commuting twists. As is often the case the results are easily predictable from the discrete case, where direct geometric proofs are elementary. The proofs however are a different and much more involved matter. These results -and more -in the procongruence case essentially follow from Theorems 3.1 and 3.8, as well as the improvement of the latter in [START_REF] Boggi | A restricted Magnus property for profinite surface groups[END_REF] ( § §5,6). They are again not available to-date in the full profinite setting.

First an important connection between commuting twists and "nonintersecting procurves" is given by: Theorem 4.1. (cf. [START_REF] Boggi | On the procongruence completion of the Teichmüller modular groups[END_REF], Corollary 6.4). Let α 1 , α 2 , . . . , α k ∈ Ľ(S) be proloops and τ h1 α1 , τ h2 α2 , . . . , τ h k α k ∈ Γ(S) denote nontrivial powers of the associated twists (h 1 , h 2 , . . . , h k ∈ Ẑ \ {0}). Then the τ hi αi 's pairwise commute if and only if the α i 's span a simplex α ∈ ČL (S).

Moreover the centralizer ZΓ(τ h1 α1 , τ h2 α2 , . . . , τ h k α k ) ⊂ Γ(S) of this family of powers of twists coincides, up to possible permutations of the curves, with the stabilizer Γα of the simplex α for the action of Γ(S) on ČL (S).

Before the statement we used the phrase "nonintersecting procurves" with inverted commas. It shoud be understood that the fact that the curves α 1 , α 2 , . . . , α k span a simplex of ČL (S) (equivalently of Č(S)) defines them as "nonintersecting". There is no direct definition available in a profinite context. The theorem above says that nonintersection may equivalently be characterized by the commutation of any set of nontrivial powers of the associated twists.

Normalizers of finite families of commuting twists are quite close to their centralizers. Just as in the discrete case, they only differ by a possible finite group of permutations. More precisely let again α 1 , α 2 , . . . , α k ∈ Ľ(S) span a (k -1)-simplex α ∈ ČL (S), let h = {h 1 , h 2 , . . . , h k } denote a k-tuple of nonzero profinite integers, and let τ h α = {τ h1 α1 , τ h2 α2 , . . . , τ h k α k } be the corresponding family of powers of twists. Finally, let G α,h ⊂ Γ(S) denote the closed free abelian group spanned by the components of τ h α . We will abbreviate this to G α if h i = 1 for all i = 1, . . . , k. With these pieces of notation we have: Theorem 4.2. (cf. [START_REF] Boggi | On the procongruence completion of the Teichmüller modular groups[END_REF], Theorem 6.6). The normalizer NΓ(G α,h ) ⊂ Γ(S) coincides with the stabilizer Γα of the simplex α for the action of Γ(S) on ČL (S).

We refer the reader to [START_REF] Boggi | A restricted Magnus property for profinite surface groups[END_REF], Corollary 6.2 for a strengthening of Theorems 4.1 and 4.2 to the analogous description of the centralizer and normalizer of a single multitwist, as a corollary of the following result. Let α = {α 1 , α 2 , . . . , α k } ∈ Č(S) k-1 and β = {β 1 , β 2 , . . . , β } ∈ Č(S) -1 be two simplices, h = {h 1 , h 2 , . . . , h k } and i = {i 1 , i 2 , . . . , i } two sets of nonzero profinite integers. Then we have: Theorem 4.3. (cf. [START_REF] Boggi | A restricted Magnus property for profinite surface groups[END_REF], Theorem 6.1). The equality of the products

τ h1 α1 τ h2 α2 • • • τ h k α k = τ i1 β1 τ i2 β2 • • • τ i β holds in Γ(S) if and only if k = l, α = β and h = i.
It is both telling and useful to rephrase the statements above in a more topological and intrinsic fashion. Recall that for a simplex α ∈ Č(S)( ČL (S)), we denote by G α ⊂ Γ(S) the closed abelian subgroup generated by the twists along the multicurves defining the vertices of α. We actually already proved the following ii) The intersection of the groups G α and G β is given by

G α ∩ G β = G α∩β . In particular, G α ∩ G β is open in α (resp. β) if and only if α ⊂ β (resp. α ⊂ β).
Given the above it is now easy and useful to manufacture yet another representation of the congruence curves complex, which we call ČT (S) (T for twist). Let us start from the discrete situation. Then we have the curve complex C(S) and two incarnations or representations of it, C L (S) and C G (S), respectively by means of curves and conjugacy classes of cyclic subgroups of π = π 1 (S). All three are isomorphic and it is also fairly easy to prove the discrete version of Proposition 4.4 above. In particular, for a simplex α ∈ C(S), G α ⊂ Γ(S) denotes the free abelian group generated by the commuting twists along the curves defining the vertices of α and in the statement of the discrete analog of Proposition 4.4 one should read "open" as "finite index". Let now G(Γ) denote the (discrete) poset of all the subgroups of Γ = Γ(S). There is a natural map

C(S) → G(Γ)
which to a simplex α associates the group G α . It is injective by the discrete analog of ii) in Proposition 4.4 and the image has a natural structure of simplicial complex induced by that of C(S). We call this image C T (S); it is (tautologically) isomorphic to C(S) and realizes this complex inside G(Γ), which is equipped with a natural action of Γ by conjugation.

We now return to the procongruence setting, adding in a useful refinement; namely we would like to work "virtually" in the sense of group theory, that is up to considering open subgroups. (This is of course doable, mutatis mutandis, in the discrete case as well.) The first observation is that the closed subgroups of a profinite group have a natural structure of profinite (po)set. Sticking to our specific case, with Γ = Γ(S), we define G( Γ), as the set of closed subgroups of Γ, which can be written as

G( Γ) = lim ←- λ∈Λ G(Γ/Γ λ ),
exhibiting it as a profinite set. Here Γ λ runs through the normal congruence subgroups of Γ and G(Γ/Γ λ ) denotes the finite set of the subgroups of the finite group Γ/Γ λ . We also have an action of Γ on G( Γ) by conjugation, as well as a (Γ -Γ)-equivariant map G(Γ) → G( Γ) sending a subgroup of Γ to its closure in Γ.

On the other hand we define a weight function on procurves w : Ľ(S) → Z * + , with values in the strictly positive integers, requiring that it be Γ-invariant. Since the Γ-orbits of Ľ(S) are in one-to-one correspondence with the types of (ordinary) loops on S, there only remains to assign an arbitrary (strictly positive) integer to each of the finitely many types. In a more geometric or modular way, this is tantamount to assigning such an integer to every irreducible component of the divisor at infinity of the stable compactification of M(S).

Given a weight function w we now consider the map C(S) → G( Γ) which sends a discrete simplex α = {α 1 , . . . , α k } to the closed abelian subgroup generated (as a Ẑ-module) by the τ w(αi) αi (i = 1, . . . , k). We then take the closure of the image of C(S) and call it ČT ,w (S) ⊂ G( Γ); it is equipped again with a structure of profinite simplicial complex and an action of Γ(S). In a slightly more intrinsic fashion this amounts to considering the discrete weighted complex C T ,w (S) ⊂ G(Γ) as mentioned above and map it to G( Γ) via the natural map G(Γ) → G( Γ). The closure of the image is by definition ČT ,w (S). This being said, the following result and its easy proof should not come as a surprise (compare [START_REF] Boggi | On the procongruence completion of the Teichmüller modular groups[END_REF], Proposition 6.8):

Theorem 4.5. Let w be a weight function as above. There is a natural Γ-equivariant isomorphism Č(S) ČL (S) → ČT ,w (S).

Moereover the images ČT ,w (S) ⊂ G( Γ) for varying w are naturally isomorphic.

Proof. The map in the statement exists by universality and is onto by the usual argument: C(S) is dense in Č(S) which is compact so that its image is closed. It is injective by Proposition 4.4. Note that the weight function w admits a unique extension from L(S) to Ľ(S) since any proloop belongs to the Γ-orbit of a discrete loop and the Γ-action is type preserving. Finally i) in Proposition 4.4 shows that we can work virtually. Namely consider the trivial weight function w 0 , assigning weight 1 to every element of Ľ(S) and write ČT (S) = ČT ,w0 (S). Then there is a canonical isomorphism ČT ,w (S) ČT ; it is defined simply by mapping every power τ w(ai) αi

(i = 1 . . . , k) to the twist τ αi itself.
The introduction of the weight function simply provides an explicit basis of open subgroups of the groups G α , just as with the groups G α,h . One can again rephrase the above in a more intrinsic fashion as follows (compare [START_REF] Boggi | On the procongruence completion of the Teichmüller modular groups[END_REF], Theorem 6.9). Let Γλ ⊂ Γ be a normal open subgroup of Γ, equivalently the closure in Γ of a normal conguence subgroup Γ λ ⊂ Γ. We can form the profinite set G( Γλ ) of the closed subgroups of Γλ and there is a natural map G( Γ) → G( Γλ ) defined by mapping each subgroup G ⊂ Γ to the intersection G ∩ Γλ . Now consider the prosimplicial complex ČT (S) = ČT ,w0 (S) as above, which on the vertices is defined simply by mapping any (pro)loop γ ∈ Ľ(S) to the associated (pro)twist τ γ ∈ Γ(S). Then the images of ČT (S) in G( Γλ ) for varying λ ∈ Λ are naturally isomorphic; in other words, the map ČT (S) → G( Γλ ) is injective for every λ ∈ Λ. The proof amounts to a translation of the above.

Let us briefly summarize where we stand. Given a hyperbolic surface S, we first defined the congruence curve complex Č(S) by completing the usual discrete version C(S), using the action of the Teichmüller group. We now have at our disposal three other realizations of Č(S), namely ČL (S) and ČG (S) which are both constructed by using the fundamental group π = π 1 (S) of the surface and finally ČT (S) which uses Γ(S), the fundamental group of the moduli stack M(S). All four appear as the completion or closure of natural discrete versions; referring to Grothendieck's manuscript Longue marche à travers la théorie de Galois, they are equipped with natural "discretifications". They are also provided with a natural action of Γ(S) extending the actions of Γ(S) on the respective isomorphic discrete versions. All four are isomorphic; moreover the isomorphisms are Γ(S)-equivariant and "natural" in the sense that once more they extend the obvious or say, geometric isomorphisms between the discrete versions (they are also natural with respect to varying S). Finally there is a dictionary between the graph or complex theoretic side and the group theoretic side, again extending the elementary discrete, geometric dictionary.

Procongruence complexes : structure and reconstruction

This section revolves around three results. First we show that the complexes Č(S g,n ) are not isomorphic for different values of the type (g, n) (the latter being well-defined thanks to the results of §3 above ; see especially Proposition 3.7) except for a few low dimensional exceptions which already occur in the discrete setting. Indeed our result parallels the analogous one in the discrete case although the proof significantly departs from the discrete one. We refer especially to [START_REF] Luo | Automorphisms of the complex of curves[END_REF] for the proof in the discrete setting, including some more references and background. Note that this already parallels a classical nonisomorphism result for Teichmüller spaces, due to D.B.Patterson (see also [START_REF] Earle | On isometries between Teichmüller spaces[END_REF]). Next we elucidate the structure of the procongruence complex of curves Č(S), or rather its close cousins Č * (S) and ČP (S), in a fashion which again parallels the discrete setting as discussed in §2 above. Finally we prove a reconstruction result in the procongruence case, on the model of Theorem 2.10 above for the discrete complexes. 5.1. Isomorphisms and non isomorphisms among the congruence curve complexes. Our first result reads: Theorem 5.1. Let S = S g,n and S = S g ,n be two connected hyperbolic surfaces of different types (g, n) and (g , n ). Then the procongruence complexes Č(S) and Č(S ) are not isomorphic, except for the following exceptional cases: Č(S 1,1 ) Č(S 0,4 ), Č(S 1,2 ) Č(S 0,5 ) and Č(S 2,0 ) Č(S 0,6 ).

Before going into the proof proper, let us dispose of the low dimensional exceptions. We mentioned the one dimensional cases for the sake of completeness only; the isomorphism is then tautological, provided that C(S 1,1 ) and C(S 0,4 ) are redefined properly, as explained in §2 (see also § §A.7, 8). The two and three dimensional cases stem directly from the exceptional discrete cases (see e.g. [START_REF] Luo | Automorphisms of the complex of curves[END_REF]). One has C(S 1,2 ) C(S 0,5 ). Then Č(S 1,2 ) (resp. Č(S 0,5 )) is the completion of that complex with respect to the action of Γ 1, [START_REF] Artin | Etale Homotopy[END_REF] (resp. Γ 0, [START_REF] Boggi | Galois coverings of moduli spaces of curves and loci of curves with symmetry[END_REF] ). However, Γ 1, [START_REF] Artin | Etale Homotopy[END_REF] acts via the quotient by its center Γ 1, [START_REF] Artin | Etale Homotopy[END_REF] /Z (Z = Z(Γ 1, [START_REF] Artin | Etale Homotopy[END_REF] ) Z/2) and we have an inclusion Γ 1, [START_REF] Artin | Etale Homotopy[END_REF] /Z ⊂ Γ 0, [START_REF] Boggi | Galois coverings of moduli spaces of curves and loci of curves with symmetry[END_REF] where Γ 1, [START_REF] Artin | Etale Homotopy[END_REF] /Z can be identified with the stabilizer of one of the 5 marked points, so has finite index (= 5) in Γ 0, [START_REF] Boggi | Galois coverings of moduli spaces of curves and loci of curves with symmetry[END_REF] . This implies that Č(S 1,2 ) Č(S 0,5 ). The last case is analogous. In order to prove Theorem 5.1 we first of all have to drastically reduce the number of possible isomorphisms between two complexes Č(S g,n ) and Č(S g ,n ) for different types (g, n) and (g , n ). This we do by introducing two invariants. The first one is the dimension of the complex, d g,n = dim(C(S g,n )) = 3g -3 + n. It is indeed invariant under completion, since C(S g,n ) injects densely into its completion Č(S g,n ). We will then introduce another invariant, or rather two closely connected ones, which will require some preliminary lemmas. This departs from the discrete setting, where the cohomological dimension of the complex C(S g,n ) provides a second invariant (see [START_REF] Luo | Automorphisms of the complex of curves[END_REF] and item ii) in Remark 5.1 below).

The first lemma-definition introduces a useful invariant in the discrete case, which will subsequently be shown to survive completion. Denote by L C (σ) the link of the simplex σ in a simplicial complex C. We define a graph L - C (σ), the dual link of σ, as follows: the set of vertices is the same as that of L C (σ), and we add an edge joining two vertices in L - C (σ) if and only if these are not joined by an edge in L C (σ). Also and following [START_REF] Luo | Automorphisms of the complex of curves[END_REF], we say that a simple loop on S (or rather an element of L(S)) is of boundary type if it bounds a subsurface of type (0, 3). The following lemma is immediate, after recalling that C(S 0,3 ) is empty: Lemma 5.2. Let α ∈ L(S) be a simple closed curve on S connected hyperbolic. The dual link L - C(S) (α) is nonempty if S is different from S 1,1 (α nonseparating) and S 0,4 (α separating and of boundary type). If nonempty, the dual link L - C(S) (α) is connected if and only if α is either nonseparating or of boundary type.

The key property to be used in the sequel is a certain type of persistence upon completion. We start with a definition. For any simplicial complex C, we say that it is (finitely) chain connected if every pair of vertices can be joined by a finite chain of edges in C. Note that this only depends on the 1-skeleton of C, and indeed below we work only with graphs. Note also that this is a combinatorial rather than topological property and indeed part of the argument below is combinatorial, independent of the underlying profinite topology. Here is the main invariance lemma: Proof. Here we put the word "chain" between brackets in the discrete case because it is clear that L - C(S) (α) is chain connected if and only if it is connected for the usual topology.

Let us first express the fact that two procurves α, β ∈ Ľ(S) are disjoint, in a concrete, combinatorial way. They are represented by coherent systems (α λ ) λ∈Λ and (β λ ) λ∈Λ where Λ denotes as usual the inverse system of the congruence levels and α λ , β λ ∈ L(S). One can project α λ (resp. β λ ) to α λ ∈ L λ (S) = L(S)/Γ λ = (C(S)/Γ λ ) (0) = C λ (S) (0) (idem β λ ). For λ large enough (Γ λ small enough), the group Γ λ ⊂ Γ(S) acts simplicially on C(S). So we can choose α λ and β λ to be adjacent in C(S) (that is, connected by an edge), and the projections α λ and β λ will then be adjacent in the quotient complex C λ (S). We thus conclude that α, β ∈ Ľ(S) are adjacent, that is are joined by an edge in Č(S) (1) , if and only if they can be represented by coherent systems α λ , β λ ∈ L(S) such that α λ and β λ are adjacent (that is, are disjoint) in C(S) for λ large enough (they may coincide for a finite number of λ's). If one of the two curves is discrete, say β ∈ L(S), the same holds with β λ = β for every λ ∈ Λ. Conversely two curves α, β ∈ Ľ(S) are not adjacent in Č(S), or say have nontrivial intersection, if and only if for every representatives (α λ ) λ∈Λ and (β λ ) λ∈Λ , α λ ∈ L(S) and β λ ∈ L(S) intersect non trivially for λ large enough.

Going back to the proof of the lemma, let us first assume that the dual link L - Č(S) (α) is chain connected in the congruence completion. By the previous lemma we have to show that α is either nonseparating or of boundary type. Assume the contrary, that is α separating not of boundarytype; we will show that it leads to a contradiction. By our assumption there exist then two loops β, γ ∈ L(S) lying in different connected components of the slit surface S α . They determine two vertices of L - Č(S) (α) and since this complex is chain connected there exists a finite chain of (pro)loops ζ i ∈ Ľ(S) (j = 0, . . . , k) connecting β and γ in L - Č(S) (α):

ζ 0 = β, ζ k = γ.
By definition this means that ζ j is not adjacent to ζ j+1 for j = 0, 1, . . . , k -1. On the other hand α is adjacent to ζ j for all j = 0, 1, . . . , k, so there are defining families (ζ j,λ ) λ∈Λ with ζ j,λ ∈ L(S) disjoint from α ∈ L(S) for all j and for λ large enough. But now since ζ j and ζ j+1 are not adjacent, we find that ζ j,λ and ζ j+1,λ intersect nontrivially for λ large enough. In particular for any such λ the chain (ζ i,λ ) j∈(0,k) is connected and joins the two connected components of S α , a contradiction.

Conversely, assume that L - C(S) (α) is chain connected and let β, γ be two vertices of L - Č(S) (α). We want to prove that there is a finite path in L - Č(S) (α) connecting them. If { β, γ} is an edge of L - Č(S) (α) there is nothing to prove. If not the three pairs of vertices of the triplets {α, β, γ} are edges of Č(S) and the latter being a flag complex, {α, β, γ} forms a triangle (a 2-simplex), that is it belongs to Č(S) (2) . Now there exists g ∈ Γ = Γ(S) such that g • {α, β, γ} = {α, β, γ} ∈ C(S) (2) for some loops β, γ ∈ L(S); indeed we may -and did -choose g ∈ Γα ⊂ Γ, the stabilizer of α. By assumption we can now find a path between β and γ in L - C(S) (α) and pull it back via g -1 to a path between β and γ in L - Č(S) (α), which completes the proof. Note that we have actually been using a basic property of the procongruence topology, namely that it is inherited by a subsurface obtained by cutting a given surface S along a multi curve. Here the closure Γα in Γ of the discrete stabilizer along the curve α is isomorphic to its congruence completion Γα , that is the completion of the modular group Γ(S α ) of the surface slit along α. This important property will be further explicited and proved in general in Proposition 5.8 below.

Next we define two closely connected invariants and explicitly compute them in the discrete case before showing that they survive after completion. Given a connected hyperbolic surface S, we let Sep(S) denote the maximal number of pairwise disjoint separating curves not of boundary type on the surface S. Here and below "disjoint curves" means as usual "disjoint elements of L(S)", that is "free isotopy classes of simple closed curves with disjoint representatives". We also denote by N Sep(S) the maximal number of disjoint curves which are either nonseparating or of boundary type on S. It is fairly easy to compute these numbers explicitly. This is taken care of by the following counting lemma: Lemma 5.4. We have

Sep(S g,n ) =    max(n -5, 0), if g = 0; max(n -2, 0), if g = 1; 2g + n -3, if g ≥ 2. N Sep(S g,n ) = n 2 , if g = 0; 3g + n -3, if g ≥ 1.
Proof. Any maximal set of disjoint curves on S 0,n has n -3 elements, among which at least two are of boundary type. Hence Sep(S 0,n ) ≤ n -5. In order to check equality, define inductively a system of curves by starting with a boundary type curve α 1 , adjoining iteratively a new curve α i surrounding α i-1 and a new boundary component (or marked point).

A separating curves on S 1,n bounds a copy of S 1,k , k ≤ n. Thus there are at most n -1 disjoint such curves, among which at least one is of boundary type. This yields Sep(S 1,n ) ≤ n -2. Again equality is attained by an obvious variant of the system of curves constructed above in genus 0.

Note now that the maximal number of pairwise disjoint separating curves on S g,n is 2g + n -3 for any (g, n), so that Sep(S g,n ) ≤ 2g + n -3. For g ≥ 2 it is immediate to construct a system with exactly this number of disjoint separating curves, none of which is of boundary type ; the equality follows.

Moving to the computation of N Sep(S g,n ), if g = 0, all curves are separating and one can arrange as many curves of boundary type on S 0,n as there are pairs of boundary components (or marked points), namely at most n 2 . For g ≥ 1, a set of pairwise disjoint curves on S g,n has at most 3g -3 + n elements (= d g,n + 1) so that N Sep(S g,n ) ≤ 3g + n -3. Since one can construct a system with this many nonseparating curves on S g,n , equality holds true.

We now show the invariance of these numbers under isomorphisms of complexes. Here we need only consider (combinatorial) simplicial isomorphisms, without any topological requirement, that is continuity with respect to the natural profinite topology is not required in the procongruence (or possibly a more general profinite) setting, as will be the case in the next paragraph. Proof. Let α = (α 1 , α 2 , . . . , α s ) ∈ C(S) (s-1) be a simplex with every loop α i ∈ L(S) separating and not of boundary type. We assume α has maximal dimension, that is s = Sep(S). Since φ is simplicial, φ(α) is a simplex of Č(S ). As usual there exists a discrete simplex in the orbit of φ(α), that is a g ∈ Γ(S ) such that g • φ(α) ∈ C(S ) ; call this discrete simplex β = (β 1 , β 2 , . . . , β s ) ∈ C(S ) (s-1) .

Since both φ and the Γ(S )-action are simplicial, for every i = 1, 2, . . . , s, the link L Č(S) (α i ) and dual link L - Č(S) (α i ) are combinatorially isomorphic to L Č(S) (β i ) and L - Č(S) (β i ) respectively. Now since L - Č(S) (α i ) is not chain connected we derive that L - Č(S) (β i ) is not chain connected either and hence by Lemma 5.3 the curves β i are separating not of boundary type. This implies that Sep(S) ≤ Sep(S ). The reverse inequality follows by symmetry, that is by considering the inverse isomorphism φ -1 .

The proof for N Sep is completely similar.

We are now finally in a position to complete the proof of Theorem 5.1. Assume there exists a simplicial isomorphism φ : Č(S) → Č(S ), where S = S g,n and S = S g ,n are of types (g, n) and (g , n ) respectively. It follows that C(S) and C(S ) satisfy the equalities :

dim(S) = dim(S ), Sep(S) = Sep(S ), N Sep(S) = N Sep(S ).
Straightforward bookkeeping using Lemma 5.4 shows that if the two types are different, the only possible isomorphisms occur for:

(1) g = 2, n ≥ 0, g = 0, n = n + 6 and n = n+6 2 , so that n = 0. In this case (g, n) = (2, 0) and (g , n ) = (0, 6).

(2) g = 1, n ≥ 1, g = 0, n = n + 3 and n = n+3 2 , so that n ∈ {2, 3}. Then (g, n) = (1, 2), (g , n ) = (0, 5) or (g, n) = (1, 3), (g , n ) = (0, 6).

(3) g = 2, n ≥ 0 and g = 1, n = n + 3.

In the first two cases we have to exclude a possible isomorphism betwee Č(S 1,3 ) and Č(S 0,6 ). Assume such an isomorphism φ : Č(S 1,3 ) → Č(S 0,6 ) does exist. Let α and γ be nonseparating curves on S 1,3 and β of boundary type, such that {α, β, γ} ∈ C(S 1,3 ) (2) form a (discrete) triangle. Next pick g ∈ Γ(S 0,6 ) such that {α , β , γ } = g • φ({α, β, γ}) ∈ C(S 0,6 ) is a discrete triangle (both Γ((S 0,6 ) and φ act componentwise).

Since the dual links of α and α = g • φ(α) are isomorphic (ditto for β and γ), it follows that α , β and γ are disjoint loops of boundary type on S 0,6 . There then exists a diffeomorphism of S 0,6 swapping α and β while fixing γ . Let h denote its class in the extended maping class group of S 0,6 ) (it may not preserve the orientation) and let H = φ -1 g -1 hgφ ; it is an automorphism of Č(S 1,3 ), exchanging α and β while keeping γ fixed.

Let now δ be a loop disjoint from α and β, intersecting γ in two points such that δ separates S 1,3 into two components S 1,1 containing α and S 0,4 containing β. Lemma 5.3 shows that the dual link L - Č(S1,3) (δ) is disconnected. Consider the preimage {α, β, H -1 (δ)} ∈ Č(S 1,3 ) (2) of the simplex {β, α, δ} ∈ C(S 1,3 ) (2) via H and let f ∈ Γ(S 1,3 ) be such that f • {α, β, H -1 (δ)} ∈ C(S 1,3 ) (2) is a discrete triangle. The curves α and β form a cut pair, dividing S 1,3 into two components which are copies of S 1,1 and S 0,4 respectively. One can take f in the stabilizer of α and β, indeed in a group of type Γ0,4 = Γ(S 0,4 ) (which is profree on two generators) because f • H -1 (δ) lies in the copy of S 0,4 . The curve f • H -1 (δ) is either nonseparating as a curve on S 1,3 or of boundary type. In both cases its dual link in the completed complex, L - Č(S1,3) (f • H -1 (δ)) is chain connected. On the other hand it should be isomorphic to L - Č(S1,3) (δ), which is disconnected. This contradiction proves that Č(S 1,3 ) and Č(S 0,6 ) are not isomorphic; of course Č(S 1,3 ) and Č(S 2,0 ) are not isomorphic either, since Č(S 0,6 ) and Č(S 2,0 ) are indeed isomorphic. Turning to the last case (3) and a putative isomorphism φ : Č(S 2,n ) → Č(S 1,n+3 ) we consider φ(α) where α is nonseparating on S = S 2,n . Its image is a vertex of Č(S ) (S = S 1,n+3 ) which can be mapped to a discrete loop by some g ∈ Γ(S ). This curve β = g • φ(α) ∈ L(S ) has a well-defined type when considered in Ľ(S ) by Proposition 3.3 and the links and dual links of α and β are isomorphic. Thus β is either separating or of boundary type. Moreover the links of α in Č(S 2,n ) and of β in Č(S 1,n+3 ) should be isomorphic. But the first link is isomorphic to Č(S 1,n+2 ) and the second one to either Č(S 0,n+5 ), if β is nonseparating or to Č(S 0,n+2 ) if β is of boundary type. Using what we did above in case (2) we conclude that β should be nonseparating and that n ∈ {0, 1}. Thus either (g, n) = (2, 0), (g , n ) = (1, 3) or (g, n) = (2, 1), (g , n ) = [START_REF] Anderson | Exactness properties of profinite completion functors[END_REF][START_REF] Boggi | On the procongruence completion of the Teichmüller modular groups[END_REF]. Finally, an isomorphism between Č(S 1,4 ) and Č(S 2,1 ) would send a nonseparating curve α on S 1,4 to a nonseparating curve on S 2,1 with isomorphic links. However these are isomorphic to Č(S 0,6 ) and Č(S 1,3 ), respectively and it was shown above (case (2)), that these complexes are not isomorphic. This completes the proof of the theorem. Remark 5.1. i) As can be readily checked Theorem 5.1 is actually valid, with the same proof, for any residually finite completion of the curve complexes C(S g,n ). Completing C(S) with respect to a quotient Γ(S) of the full profinite completion Γ(S), it amounts to requiring (see [START_REF] Boggi | Profinite Teichmüller theory[END_REF], Prop. 5.1) that Γ(S) be residually finite (i.e. the natural map Γ(S) → Γ(S) should be into). In particular this is the case of any completion which is finer than the congruence completion. ii) By a famous result of Harer-Ivanov, the curve complex C(S) is homotopically equivalent to a wedge of spheres of dimension h(S). The value of h(S g,n ), which is also the cohomological dimension of C(S g,n ), is explicit (see e.g. [START_REF] Luo | Automorphisms of the complex of curves[END_REF]) and provides a second invariant (after the dimension) in the discrete case. However in the complete case we could not use it for a reason which perhaps deserves to be mentioned. One knows that H q (C(S), Q) vanishes for all q = 0, h (h = h(S)); the same is true of every finite quotient H q (C(S)/Γ λ , Q) (Γ λ normal cofinite in Γ(S)). But we were not able to show that H h (C(S)/Γ λ , Q) does not vanish. In fact one would like to show this for some (any) value of λ and in particular one can take λ large enough (Γ λ small enough) so that all the components of the boundary ∂M λ (S) of the associated level are smooth. The question is whether the combinatorics of these components contributes to the cohomology of the simplicial variety ∂M λ (S). Note that H h (C(S)/Γ λ , Q) injects into the Hodge weight 0 part of the rational cohomology of ∂M λ (S). Is W 0 H h (∂M λ (S), Q) nontrivial for some λ, in particular for Γ λ = Γ(S), M λ (S) = M(S)) ?

5.2.

Local structure and reconstruction of congruence graphs. Our next objective is the profinite analog of Theorem 2.10, which is interesting for its own sake and will be used in the next section, much as was done in §2 in the discrete setting, in order to start exploring the continuous automorphisms of the congruence complexes. In this subsection we will deal almost exclusively with the graphs C * (S) and C P (S) and their congruence completions, as they carry most of the relevant information. We refer to § §A.5, 7, 8, 9 for the basic definitions. Note that we will not make use of the isomorphism theorem in what follows, and for good reasons since we have not shown any result of that type pertaining to these graphs. It could however be interesting to state and prove such results.

We start from a surface S which is not assumed to be connected but is such that each of its finitely many connected component S i is hyperbolic (S = i∈I S i ). We define Γ(S) = i∈I Γ(S i ), the colored modular group, and let each Γ(S i ) act naturally on C * (S i ) and C P (S i ) so as to extend definitions to the non connected situation (see also §A.9). Finally we deal with the congruence completions Č * (S) and ČP (S) by completing the modular groups Γ(S i ) of the connected components. Note that possible permutations of the pieces have no effect on completions, since they generate a finite group; in other words, the colored modular group has finite index in the full modular group.

If S is connected with d(S) = 0 i.e. is a trinion (a.k.a a pair of pants), Č * (S) = ČP (S) is empty or conventionally reduced to a point and coincides with its discrete version. If S is connected with d(S) = 1, Č * (S) = Ǧ and ČP (S) = F , where the completion can be taken with respect to the natural action of Γ 0,4 ( F 2 ), which has finite index in Γ 1,1 (see § §A.7,8) Although for resaon of coherence we use the notation for the congruence completion, here it does not differ from the full profinite completion. Recall that more generally the congruence conjecture holds for types (g, n) with g = 0, 1, 2, and n arbitrary. If d(S) > 1, Č * (S) identifies with the 1-skeleton of the dual of Č(S) but we are aiming at a direct description, actually valid in all dimensions d(S) ≥ 0. There is a natural action of Γ(S) on Č * (S) and ČP (S) and as usual, one can describe their common set of vertices (denoted V (S)) as a finite disjoint union v∈F Γ•v of Γ-orbits of discrete vertices v ∈ V (S). Each v ∈ V (S) represents a discrete maximal multicurve (a.k.a. a pants decomposition) of S and the finite set F enumerate the types (Γ-orbits) of such deompositions. The set Ě(S) of edges of Č * (S) can be described as follows:

Lemma 5.6. The vertices v, w ∈ Č * (S) are joined by an edge if and only the corresponding maximal multicurves differ by exactly one component up to relabeling.

Proof. The statement should be interpreted as follows. Write v = (α 1 , . . . , α k ) (resp. w = (β 1 , . . . , β k )) where the α i 's and β j 's are (pro)curves and k = d(S) + 1). One could assume that either v or w corresponds to a discrete pants decomposition but that does not really help. The claim is that the condition for v and w to be joined by an edge in Č * (S) is the exact analog of what happens in the discrete case.

The "if" part of the statement is clear and we have to show the "only if" part. In order to do this, let v = lim ←-λ∈Λ v λ , w = lim ←-λ∈Λ w λ where λ ∈ Λ belongs to the set of congruence levels (here we may assume S connected for simplicity) and v λ , w λ ∈ C λ * = C * (S)/Γ λ . We can write v λ = (α λ i ), w λ = (β λ i ) where the α λ i and β λ i represent Γ λ -orbits of curves (i.e. they lie in L(S)/Γ λ ). Moreover since v and w are joined by an edge in Č * , there exist discrete pants decompositions (A λ i ), (B λ j ) in C * which project to v λ and w λ repectively and are joined by an edge in C * . So (A λ i ) and (B λ j ) differ by at most one curve, after relabeling. For any λ ∈ Λ consider the label (in {1, . . . , k}) of the curve in the family (A λ i ) which does not occur in (B λ j ) (if they coincide pick any label). This may depend also on the chosen lifts of v λ and w λ but that does not matter. Now consider a cofinal sequence in Λ and choose a label which occurs infinitely often in the above construction. One finds that v and w can indeed be represented by multi(pro)curves (α i ) and (β i ) which coincide except for the entry in v corresponding to that label.

We are heading toward a statement and proof of the procongruence analog of Theorem 2.10, which deals with the graph Č * (S). We take up the setting and notation of the beginning or §2.2, starting with a hyperbolic surface S. We assume that all connected components of S have the same modular dimension, which we denote d(S), and that d(S) > 0. For a multicurve σ ∈ C(S), S σ denotes, as in §2, the surface S slit along the multicurve σ. We will first show that given such a multicurve, there is a natural embedding of the procongruence curve graph Č * (S σ ) into Č * (S) and that it is equivariant with respect of the actions of the attending modular groups Γ(S σ ) and Γ(S). This in essence is not new but it does embody an essential property of the proconguence topology, which we summarize in the following geometric lemma: Lemma 5.7. Let S be as above, σ ∈ C(S) a multicurve, S σ the surface with boundary obtained by cutting S along σ. Then every unramified Galois cover of S σ is dominated by a Galois cover of S.

Proof. To be sure, the lemma asserts that one can find a Galois cover of S which restricts to a cover of the multicurve σ, does not permute the pieces of S σ , and dominates the given cover of S σ as a surface with boundary. The proof is in fact elementary. By an immediate induction one restricts to the case of a single curve σ = {α} and this case is dealt with in [START_REF] Boggi | Profinite Teichmüller theory[END_REF], Lemmas 6.7 (α non separating) and 6.8 (α separating).

We insist that this elementary and relatively easy lemma is nonetheless a key point. In essence it says that the procongruence topology transfers nicely when cutting along a multicurve, producing subsurfaces of the ambient surface; see Proposition 5.8 below. The analog in the full profinite case, where one has to work with covers of moduli stacks and not just surfaces is not known.

We will now proceed to state the proposition we need in order to make good sense of the reconstruction problem in the procongruence setting. The proof is again quite easy, given the lemma above. In fact we will state the proposition for Γ(S), C * (S) and C P (S) simultaneouslys because these are the objects we have to deal with but it simply expresses again the fact that the topology induced -so to speak -on S σ by the congruence topology attached to S coincides with the congruence topology attached to S σ ; this in turn is nothing but the content of Lemma 5.7 above. Put this way it is clear that it applies "functorially" and equivariantly to a lot of objects attached to surfaces (starting with C(S), the curve complex itself), provided they display a nice behavior with respect to the operation of "cutting along multicurves". We refrain from giving a more abstract statement, but see Remark 5.2 below.

Given S and σ ∈ C(S) we have natural embeddings: Γ σ → Γ(S), C * (S σ ) → C * (S), C P (S σ ) → C P (S). In the first case Γ σ denotes the stabilizer of σ in Γ(S); for the two others see the beginning of §2.2 above. Using that the congruence completion is residually finite, we get corresponding embeddings Γ σ → Γ(S), C * (S σ ) → Č * (S) and C P (S σ ) → ČP (S). This leads to continuous embeddings of the respective closures:

Γσ → Γ(S), C * (S σ ) → Č * (S) and CP (S σ ) → ČP (S). Note that by the universality of the congruence completion the discrete embeddings into the respective completions factors through the completions of the respective sources. In other words we also have (continuous) maps: Γσ → Γ(S), Č * (S σ ) → Č * (S) and ČP (S σ ) → ČP (S). These however are not known a priori to be injective. In fact, we a priori get surjective maps Γ(S σ ) → Γσ , Č * (S σ ) → C * (S σ ) and ČP (S σ ) → CP (S σ ) stemming from the fact that the induced topology from S is a priori a quotient (i.e. at most as fine) as the congruence topology attached to S σ . Our next proposition asserts that these last maps are in fact isomorphisms: Proposition 5.8. Let S be as above, σ ∈ C(S) be a multicurve, S σ the surface S slit along σ. With the notation and construction as above, the resulting maps are all isomorphisms:

Γ(S σ ) ∼ -→ Γσ , Č * (S σ ) ∼ -→ C * (S σ ), ČP (S σ ) ∼ -→ CP (S σ ).
Proof. As mentioned above the proposition is a fairly straightforward consequence of Lemma 5.7. One only needs to transfer the information from covers of surfaces to the analog on congruence levels of the associated moduli spaces M(S) and M(S α ). This again is part of Lemmas 6.7, 6.8 in [START_REF] Boggi | Profinite Teichmüller theory[END_REF]. One could also use, in the same spirit, isomorphism results of the type Č(S) ČL (S) (see Theorem 3.1), whose goal is precisely to transfer information from the congruence levels of the moduli space M(S) to covers of the surface S itself -and back. But we have not shown or even stated such general results outside of the case of the curve complex C(S).

So in the end we get continuous embeddings: Γσ → Γ(S), Č * (S σ ) → Č * (S) and ČP (S σ ) → ČP (S) with closed, hence compact images since the sources are compact. From the first isomorphism, namely Γσ Γ(S σ ), we conclude that in the last two cases the maps are equivariant with respect to the action of Γ(S σ ) and Γ(S) on the sources and targets respectively. We remark that in the above we have been a little sloppy about boundary curves, not always distinguishing very carefully between a surface with or without boundary. In fact we have left it to the reader to straighten out some details for her/himself. Remark 5.2. i) As anticipated in Grothendieck's Esquisse, there exists an underlying beautiful "dictionary" between objects of a priori very different natures, from topology to arithmetic through hyperbolic, conformal, complex or algebraic geometry. For instance one can -should -consider the completed stack M(S), or more generally Mλ (S) as a simplicial or stratified object where the strata are enumerated by C(S)/Γ(S) (resp. C(S)/Γ λ (S)). Note that formally the generic stratum M(S) corresponds to S = S ∅ , that is to ∅ ∈ C(S) (-1) . In essence, Proposition 5.8 says that the congruence completion respects this simplicial character of the stably completed moduli stacks of curves.

ii) The analog of Proposition 5.8 is not known in the full profinite case and in fact its validity is equivalent to that of the congruence subgroup conjecture (this is the case for several statements in § §3, 4, 5). Indeed, assuming it holds true one can prove the conjecture working by induction on the the modular dimension d(S) and using equivariant spectral sequences as in [START_REF] Boggi | Profinite Teichmüller theory[END_REF], §6.

We now turn to the reconstruction problem. We first note that one can view ČP (S) ⊂ Č * (S) as a closed subgraph with the identical set V (S) of vertices and a set ĚP (S) ⊂ Ě(S) of edges. Indeed consider the natural injections C P (S) → C * (S) and C * (S) → Č * (S); by composition we get an equally natural injection C P (S) → Č * (S); taking the closure of C P (S) inside Č * (S) yields ČP (S) as should be clear from the above. Alternatively the injection of C P (S) into Č * (S) factors through ČP (S) by universality and the resulting map ČP (S) → Č * (S) is injective. Yet it is not so easy to give a description of ČP (S) inside Č * (S) or equivalently of ĚP (S) as a subset of Ě(S). We insist on that matter because it will turn out that, modulo reconstruction of the whole of Č(S) from the graph Č * (S) (Corollary 5.14 below) and the rigidity of ČP (S) (Theorem 7.1 below), we are getting quite close to the actual root of Grothendieck-Teichmüller theory in this profinite topological ("nonlinear") setting. In particular it will evolve (elsewhere) that the set of injective morphisms j : ČP (S) → Č * (S) is a close profinite analog of the variety of associators introduced by V.G.Drinfel'd in the prounipotent case. Here, however, we restrict attention to the "natural" injection or inclusion ČP (S) ⊂ Č * (S), or equivalently ĚP (S) ⊂ Ě(S). We are first aiming at a better understanding of the local structure of Č * (S) and ČP (S), more precisely at making sense and proving an analog of Lemma 2.7. We start with: Lemma 5.9. Let S be connected hyperbolic and let σ ∈ C(S) be a multicurve which is not maximal. Then if g ∈ Γ(S) stabilizes the subcomplex Č * (S σ ) of Č * (S), i.e. g( Č * (S σ )) = Č * (S σ ), it stabilizes σ, i.e. g(σ) = σ.

Proof. Assume that g(σ) is different from σ. We want to show that there exists a proloop γ ∈ Ľ(S) which is disjoint from σ but not from g(σ). First, there exists β ∈ L(S) contained in σ (i.e. β is a vertex of σ) such that g -1 (β) is not a vertex of σ. By hypothesis, for any τ ∈ C * (S σ ) we have g(τ ) ∈ Č * (S σ ). In particular, if we consider a maximal multicurve τ ∈ C * (S σ ) containing the curve β, we derive that the proloop g(β) ⊂ g(τ ) must be disjoint from σ, namely that the simplex (σ, g(β)) ∈ Č(S). Further, there exists h ∈ Γ(S) such that h(σ, g(β)) is a discrete simplex. We may choose h such that h(σ) = σ and hg(β) = α ∈ L(S) is then a simple closed curve disjoint from σ, so that (σ, α) ∈ C(S). The component of S σ containing α ∈ L(S σ ) cannot be of type (0, 3) since it contains the curve α. Thus S σ contains some simple closed curve γ which intersects geometrically α. The proloop γ = h -1 (γ) then satisfies the original requirement.

Observe now that any proloop γ as above can be completed to a maximal multicurve τ ∈ Č * (S σ ), since discrete curves have this property and h(γ) is discrete. Now τ contains the curve γ which intersects g(σ), so that g -1 (τ ) does not belong to Č * (S σ ), contradicting the assumption that g stabilizes Č * (S σ ).

For any subsurface Σ ⊂ S we may identify the (pro)graph Č * (Σ) (resp. ČP (Σ)) with a subgraph of Č * (S) (resp. ČP (S)). We can now proceed with: Lemma 5.10. Let Σ 1 and Σ 2 be two distinct subsurfaces of S of dimension 1. Then the intersection ČP (Σ 1 ) ∩ ČP (Σ 2 ) in ČP (S) is either empty or consists of a single vertex.

Proof. Let Σ j be the connected component of S σj of dimension 1, where the σ j (j = 1, 2) are codimension 1 simplices of C(S). The vertices of ČP (Σ j ) ⊂ ČP (S) are of the form (σ j , γj ), where γj ∈ Ľ(Σ j ) is a proloop on Σ j . If the intersection ČP (Σ 1 ) ∩ ČP (Σ 2 ) is non-empty there exist two such vertices which coincide, i.e. such that (σ 1 , γ1 ) = (σ 2 , γ2 ) as vertices of ČP (S). Since Σ 1 and Σ 2 are distinct we can write σ j = (σ j , δ j ) for simple closed curves δ j ∈ L(Σ j ) such that σ 1 = σ 2 , γ1 = δ 2 and δ 1 = γ2 . In particular γj ∈ L(Σ j ) is a (discrete) simple closed curve (j = 1, 2) and for fixed Σ j the common vertex is a unique discrete vertex.

Given σ ∈ C(S), we recall that S σ denotes the surface S obtained by cutting S along the curves in σ and then crushing boundary circles to punctures. There is a natural injection C P (S σ ) → C P (S), which sends the pants decomposition τ of S σ to the pants decomposition τ ∪ σ of S. This construction extends to the completions, as follows. Let σ ∈ Č(S). There exists then g ∈ Γ(S) and σ ∈ C(S) a discrete simplex such that g • σ = σ. We set then

ČP (S σ ) = g • ČP (S σ ) ⊂ ČP (S)
This is well-defined and independent on the choices involved, as the topological type of σ is well-defined. We will need the following properties of ČP (S): Lemma 5.11. i) ČP (S) is covered by the images of ČP (S σ ), where σ ∈ Č(S) is a simplex of codimension cd(σ) = 1:

ČP (S) = σ∈ Č(S), cd(σ)=1 ČP (S σ ).
ii) Given Σ a dimension 1 surface we construct the complete prograph C * (Σ) whose vertices are those of ČP (Σ). We define C * (S) as the quotient of the disjoint union C * (Σ), over all dimension 1 subsurfaces Σ of S, by the equivalence relation which identifies vertices v ∈ C * (Σ 1 ) and w ∈ C * (Σ 2 ) if their respective images under the natural embeddings ČP (Σ i ) → ČP (S) coincide. Then C * (S) is isomorphic to Č * (S). iii) Say that the simplices ρ and τ of Č(S) are compatible if for every pair of vertices v and w of ρ and σ respectively, either v = w or v and w are not joined by an edge in ČP (S). Then if ρ and τ are compatible Č * (S ρ ) ∩ Č * (S σ ) = Č * (S ρ∪σ );

otherwise the intersection is empty.

Proof. The first statement follows from its discrete counterpart and Proposition 5.8. The second item is a consequence of Proposition 5.8 along with Lemma 5.10. Then Lemma 5.10 and Lemma 5.6 imply the last claim, which is the procongruence analog of Lemma 2.6.

The procongruence analog of Lemma 2.7 is a straightforward consequence of this lemma, that is:

Proposition 5.12. The graph Č * (S) is obtained from ČP (S) by replacing every maximal copy of F inside ČP (S) by a copy of Ĝ.

Let us now proceed towards the reconstruction theorem, starting however with a discussion about its proper statement and meaning in the complete setting. In the discrete case the natural action of Γ(S) on C(S) translates into an action of Γ(S) on the graphs C * (S σ ), viewed as subgraphs of C * (S). For g ∈ Γ(S), σ ∈ C(S), we get the following equivariance formula:

g • C * (S σ ) = C * (S g•σ )
which also holds with C * replaced by C P . In particular the reconstruction principle of Corollary 2.11 respects the natural Γ-action.

In the procongruence case, using the natural action of Γ(S) on Č(S), we find that g • Č * (S σ ) is a welldefined closed subgraph of Č * (S) for g ∈ Γ(S) and σ ∈ C(S). At this point one is tempted to write down the same formula as above, replacing the objects with their respective congruence completions, that is g • Č * (S σ ) = Č * (S g•σ ) for any g ∈ Γ(S), σ ∈ Č(S). In the general case however, that is for g / ∈ Γ(S) and σ / ∈ C(S), neither side is a priori well-defined. If we pick σ ∈ C(S) a discrete simplex and g ∈ Γ(S) arbitrary, then the right-hand side can be defined by the left-hand side. Then extend the definition to any σ ∈ Č(S) using as usual the fact that the Γ(S)-orbit of any simplex in Č(S) contains a discrete representative.

One thus gets a family ( Č * (S σ )) σ∈ Č(S) of closed subgraphs of Č * (S) which is indexed by the profinite simplicial set Č(S) and is equipped with a natural simplicial action of Γ(S). These subgraphs are distinct for σ not maximal, that is Č * (S σ ) = Č * (S τ ) if and only if σ = τ ∈ Č(S). In fact in order to vindicate this assertion, it is enough to show that for any discrete σ ∈ C(S) and any g ∈ Γ(S), g • Č * (S σ ) = Č * (S σ ) if and only if g • σ = σ, which is Lemma 5.9 above. As in the discrete case, reconstructing Č(S) out of Č * (S) consists in graph theoretically detecting or characterizing the family ( Č(S σ )) σ∈ Č(S) , which can be made into a prosimplicial complex using the inclusion of curves as a boundary operator.

In what follows, for τ ∈ Č(S), one can think of Č * (S τ ) via the defining formula Č * (S τ ) = g • Č * (S σ ) for σ ∈ C(S) discrete, g ∈ Γ(S), g • σ = τ , thus avoiding making sense directly of the symbol S τ , that is "S slit along the profinite simplex τ ". Finally it may be worth pointing out the possible connection with what Grothendieck calls discrétifications in his Longue Marche à travers la théorie de Galois ( §26). Roughly speaking and to be specific, given a finitely generated residually finite group G and its profinite completion Ĝ one can consider the set of its discretifications, that is of the dense injections G → Ĝ. This can be seen as a natural extension of the notion of integral lattice or integral structure in the linear setting. These discretifications will form a torsor under a group which is not easy to capture in general but may be worth keeping in mind. In an analogous way one can view the set of dense embeddings C(S) → Č(S) as the set of integral structures on Č(S) and in our context the above seemingly formal definitions become more natural, since the group Γ(S) will act naturally on these structures ("discretifications") as well.

We can now state the procongruence version of Theorem 2.10 as:

Theorem 5.13. Let S be a connected hyperbolic surface, C ⊂ Č * (S) a subgraph which is topologically isomorphic to Č * (Σ) for a certain surface Σ and is maximal with this property. Then there exists a unique σ ∈ Č(S) such that C = Č * (S σ ).

Proof. If one wishes to stick to S σ for discrete simplices σ ∈ C(S), the assertion can be rephrased by saying that there exist σ ∈ C(S) and g ∈ Γ(S) such that C = g • Č * (S σ ). Two solutions (σ, g) and (σ , g ) satisfy

g • σ = g • σ ∈ Č(S).
As in the discrete setting, the case σ = ∅ should be included and corresponds to the full complex Č * (S). With Lemmas 5.6 and 5.11 at our disposal, the proof proceeds along the lines of the proof in the discrete case. We need only show the existence part, uniqueness being clear, as in the discrete case. The first step consists in showing that a subgraph of the form indicated in the statement is maximal. To this end, one can consider a discrete σ ∈ C(S) and prove that Č * (S σ ) is maximal in its isomorphy class. The proof follows the one in the discrete case in §2. [START_REF] Boggi | Profinite Teichmüller theory[END_REF].

Here and as in the discrete case again, it is more elegant (although not necessary) to include the case d(Σ) = 0, i.e. Σ a trinion, of type (0, 3), by declaring that the attending curve complex is reduced to a point rather than empty: C(S 0,3 ) = C * (S 0,3 ) = { * }. The case of dimension 0 is then clear: the vertices of Č * (S) correspond to maximal multicurves (not necessarily discrete). One can start induction from there, or treat the case d(Σ) = 1 independently, as in the discrete case. We do not do it in detail because the inductive argument applies to that case as well. Suffice it to say that it is still true that any triangle in Č * (S) defines a unique subsurface of dimension 1, possibly after twisting by an element of Γ(S).

Having disposed of the low-dimensional cases, we argue again by induction on k = d(Σ). So we pick k > 1, assume the statement is true for d(Σ) < k and fix an isomorphism

C ∼ -→ Č * (Σ). For σ ∈ C(S)
we then define C σ Č * (Σ σ ) ⊂ Č * (S) as in the discrete case. This time the union of the C σ 's as σ runs over the nonempty simplices of C(Σ) form a dense part of C, which is sufficient for the same argument as in the discrete case to go through. Namely in order to conclude the proof, it is enough to show that there exists a k-dimensional subsurface T ⊂ S and an element g ∈ Γ(S) such that for any (nonempty

) σ ∈ C(Σ), C σ ⊂ g • Č * (T ) ⊂ Č * (S).
We may again (as in the discrete case) restrict to |σ| = 1, i.e. to the discrete loops on Σ. To any such loop α ∈ L(Σ) we can attach by induction a subsurface S (α) ⊂ S of dimension k -1 and an element g α ∈ Γ(S) such that C α = g α • Č * (S (α) ) ⊂ Č * (S). As usual, having fixed an isomorphism C ∼ -→ Č * (Σ) we write an equality sign for the sake of simplicity.

Next we use, again as in the discrete case, the connectedness of C(Σ) which is ensured by the assumption on the dimension (k > 1). So we have to study the following situation. We consider three discrete loops α, β, and γ on Σ such that α ∩ β = β ∩ γ = ∅. We attach to them as above pairs (g α , S (α) = S ρ ), (g β , S (β) = S σ ) and (g γ , S (γ) = S τ ) for certain simplices ρ, σ, τ ∈ C(S) with |ρ| = |σ| = |τ | = d(S) -k + 1. Moreover ρ and σ (resp. σ and τ ) are compatible simplices.

As in the discrete case, the situation should be entirely determined by any two pairs of non intersecting curves on Σ, after which one can worry over a possible overdetermination. The reasoning below may appear more transparent if one recalls that a graph of the form g • Č * (S σ ) is actually determined by the profinite simplex g • σ and so depends on g only up to the subgroup of Γ(S) fixing σ, which is nothing but the centralizer of the multitwist corresponding to σ. These centralizers are determined in §4 above. So let us first examine what happens when trying to paste the data for α and β. After twisting we may assume that g α = 1 and write g β = g ∈ Γ(S). Next we know that the intersection C α ∩ C β has dimension k -2 and indeed is isomorphic to a twist of Č * (Σ α∪β ). This implies that |ρ ∩ σ| = d(S) -k and that g fixes = ρ ∩ σ, that is g ∈ Z . Writing T = S we find that S (α) = S ρ ⊂ T . Moreover, because g fixes we can find

h ∈ Γ(T ) such that C β = g • Č * (S σ ) = h • Č * (S σ ). But then, since h ∈ Γ(T ), h • Č * (S σ ) ⊂ Č * (T )
and so we get the inclusion C β ⊂ Č * (T ). Returning to our original notation, we found a k-dimensional subsurface

T ⊂ S such that C α ⊂ g α • Č * (T ), C β ⊂ g β • Č * (T )
and in fact g β = g α = g. Proceeding in the same way with the pair (β, γ) we get a possibly different pair (g , T ). Now in order to compare T and T , we use again the fact that there is a large intersection, namely that

C β ⊂ g • Č * (T ) ∩ g • Č * (T )
. This implies that one can modifysay -g in order to achieve g = g and then, because T , T and Σ are all of dimension k, one shows as in the discrete case that T = T .

We now draw a consequence of this recognition result, much as in the discrete case, before turning to the study of the automorphism groups of the congruence complexes. Indeed Theorem 5.13 yields the analog of Corollary 2.11: Corollary 5.14. For d(S) > 1, Č(S) can be reconstructed from Č * (S).

Proof. In fact, as mentioned above, one reconstructs Č(S) by considering the set of subgraphs of Č * (S) satisfying the conditions stated in Theorem 5.13, making it into a prosimplicial complex by using inclusion and deletion of curves as the face and boundary operators respectively. The theorem ensures that the resulting complex is indeed isomorphic to Č(S).

As a last item in this section and a corollary of what has been done above, we return to the issue of the possible isomorphisms between complexes of different types: Proposition 5.15. Let S = S g,n and S = S g ,n be connected hyperbolic surfaces of different types. Then: i) C * (S 1,1 ) C * (S 0,4 ), C * (S 1,2 ) C * (S 0,5 ), C * (S 2,0 ) C * (S 0,6 ) and there are no other isomorphims; ii) same as i) above in the procongruence setting, that is with C * (S) replaced with Č * (S) everywhere; iii) C P (S 1,1 ) C P (S 0,4 ) and this is the only nontrivial isomorphism between discrete pants graphs; iv) same as iii) above in the procongruence setting, that is with C P (S) replaced with ČP (S) everywhere.

Proof. Item i) holds true if we replace C * (S) by C(S) (see e.g. [START_REF] Luo | Automorphisms of the complex of curves[END_REF]). For d(S) = 1 C * (S 0,4 = C * (S 1,1 = G, where G is the complete graph on the vertices of the Farey graph F (see §A.8). If d(S) > 1, C * (S) is the 1-skeleton of the dual of C(S); conversely C(S) can be reconstructed from C * (S) by Corollary 2.11. So the cases of isomorphisms for C * (S) and for Č(S) coincide.

The reasoning for ii) is identical, using Theorem 5.1 and Corollary 5.14.

For iii), that is concerning the discrete pants graph, Lemma 2.7 says that C * (S) can be reconstructed from C P (S) (but not vice versa!) so that the cases of possible isomorphisms for C P (S) are among the possibilities C * (S). In dimension 1 we do have C * (S 1,1 ) C * (S 0,4 ) = F . In order to rule out the other two possibilities it is enough to show that C P (S 1,2 ) and C P (S 0,5 ) are not isomorphic. In fact, assume there exists an isomorphism φ : C P (S 2,0 ) ∼ -→ C P (S 0,6 ). Let then α ∈ L(S 2,0 ) be a nonseparating loop; it is mapped to a loop α = φ(α) ∈ L(S 0,6 ) which is of boundary type. Cutting the surfaces along α and α respectively, we find that φ would induce an isomorphism between C P (S 1,2 ) and C P (S 0,5 ). However this is impossible because it would also imply an isomorphism between the attending full (two dimensional) complexes. Now C P (S 0,5 ) contains pentagons whereas C P (S 1,2 ) does not (see [START_REF] Hatcher | On the Teichmüller tower of mapping class groups[END_REF]), which shows that these complexes are not isomorphic.

To iv) we use Proposition 5.12 to conclude that Č * (S) can be reconstructed from ČP (S) (again, this is not reversible) and that the possible isomorphisms are at most those which obtain for Č * (S). In dimension 1 we get ČP (S 1,1 ) ČP (S 0,4 ) = F but in dimensions 2 and 3, although the discrete graphs are not isomorphic, we cannot a priori rule out possible isomorphisms between their respective profinite analogs. Recall that the congruence conjecture holds for types (g, n) with g = 0, 1, 2 and n arbitrary, so that we may replace the procongruence by the (full) profinite completion as far as the low dimensional complexes mentioned above are concerned. By now it is easy to transpose the reduction argument in the proof of item iii) to the profinite setting. So the only moot point consists in showing that the completed complexes ĈP (S 1,2 ) and ĈP (S 0,5 ) are not isomorphic, which follows essentially by the same argument as in the discrete case. The reader will find in [START_REF] Lochak | Automorphism groups of profinite complexes of curves and the Grothendieck-Teichmüller group[END_REF] a detailed study of the "completed pentagons". Note that this non isomorphism statement is rather obvious from the viewpoint of Grothendieck-Teichmüller theory because owing to the two-level principle and the fact that ĈP (S 1,1 ) and ĈP (S 0,4 ) are indeed isomorphic, the discrepancy between ĈP (S 1,2 ) and ĈP (S 0,5 ) actually carries the whole difference between the genus 0 and the general case of Grothendieck-Teichmüller theory (see [START_REF] Hatcher | On the Teichmüller tower of mapping class groups[END_REF][START_REF] Lochak | Automorphism groups of profinite complexes of curves and the Grothendieck-Teichmüller group[END_REF]).

Automorphisms of procongruence complexes

We now start investigating the automorphisms of the procongruence complexes attached to a connected hyperbolic surface S, especially our three favorite complexes Č(S), ČP (S) and Č * (S). These are by definition limits of the quotients C λ (S) = C(S)/Γ λ over the inverse system λ ∈ Λ indexing the principal congruence subgroups of the Teichmüller modular group Γ(S). Now an interesting point is that the quotient C(S)/Γ λ (for λ large enough) can be considered in two ways, either "naively", as a finite CW-complex or, retaining more structure, as a topological stack, which here is almost the same thing as an orbifold, except that the stabilizers are not finite. Thus, working with the curve complex purely for definiteness, Č(S) can be considered as a pro-object either of the category of finite complexes or of topological stacks. In both cases endomorphisms are defined following the classical prescriptions for pro-objects of a category (see [START_REF] Artin | Etale Homotopy[END_REF], Appendix), namely as a system of of compatible maps C µ (S) → C λ (S) between finite complexes (so in particular continuous for the ordinary topology), with λ running over Λ and µ ≥ λ. An automorphism is an invertible endomorphism. When varying λ ∈ Λ, a basis of neighborhoods of the identity in the group Aut( Č(S)) is defined by those elements which induce the natural projection between the above quotients for some fixed λ, µ ∈ Λ. Note that these elementary neighborhoods are not subgroups. They endow Aut( Č(S)) with a structure of profinite group.

In this section we will consider the finite quotients as complexes, erasing in particular the information coming from the ramification. However in the next section, topological stacks will play an important role and we will return to the above in more detail. Note also that here ČP (S) and Č * (S) are prographs and we will recall below how Č(S), being a flag complex, is entirely determined by its 1-skeleton. So the first striking fact is that here we need actually deal only with profinite graphs, that is one-dimensional complexes.

This section aims at proving some basic and fundamental properties of the automorphism groups attached to the three complexes above. The reconstruction theorem above (Theorem 5.13) will play a significant role; on the one hand and much as in the discrete case, it paves the way towards some basic results, demonstrating how much of the information about the original curve complex Č(S) can be transferred to Č * (S), which we recall is nothing but the 1-skeleton of the dual of Č(S) (in modular dimension d(S) > 1). The gain is that we have a natural inclusion of profinite graphs ČP (S) → Č * (S) which actually summarizes the main part of the information we are interested in (see also above Proposition 5.12). Anticipating again, we remark that from the point of view of Grothendieck-Teichmüller theory, this reconstruction result demonstrates how the so-called "Teichmüller tower" is not really needed: at every level, that is for a given modular dimension d(S), the corresponding congruence curve complexes contain all the information coming from the lower levels. Putting this together with the "two level principle" ("principe des deux premiers étages"), will imply, as will be shown elsewhere, that one needs only consider a single, given profinite graph Č(S), with S of large enough dimension and genus, in order to investigate the automorphism group of the whole "tower", a kind of very strong stability result. 6.1. Basic results. First we state and prove explicitly a proposition which has already been alluded to, namely: Proposition 6.1. For any hyperbolic surface S, Č(S) is a flag complex. As a consequence every automorphism of the 1-skeleton can be extended to an automorphism of the full complex:

Aut( Č(S)) = Aut( Č(S) (1) ).

Proof. Recall that a flag complex is a simplicial complex such that every clique is a simplex. That is if σ = (v i ) i∈I is a finite set of vertices such that every pair of elements of I defines an edge, then σ is a simplex. This is obviously the case of the discrete complex C(S) but the preservation of this property under completion is in general a delicate question. Fortunately here we can take advantage of Theorem 4.5 (which itself constitute a highly nontrivial result), say for the trivial weight function. It then asserts that Č(S) ∼ -→ ČT (S) where the isomorphism is defined by mapping a simplex α = {α 1 , . . . , α k } ∈ Č(S) to the closed free abelian group G α = τ α1 , . . . , τ α k ⊂ Γ(S) generated by the corresponding protwists. It is then clear that this group is abelian if and only the twists are pairwise commuting which translates into the fact that ČT (S) ⊂ G( Γ(S)) is a flag complex, hence Č(S) as well. The second assertion of the proposition is an immediate corollary of the first.

So we have reduced our problem to studying the automorphisms of the graphs Č(1) (S), Č * (S) and ČP (S). Now the analog of Lemma 2.8 holds true for the congruence graphs thanks to Proposition 5.12 i.e. the procongruence analog of Lemma 2.7. So we get: Proposition 6.2. Given the injection ČP (S) → Č(S), there is a natural injection Aut( ČP (S)) → Aut( Č * (S)).

Note that we favored the word 'injection' over 'inclusion' because it has a more dynamical flavor and the set of (not necessarily natural) injections ČP (S) → Č(S) will play a leading role in topological Grothendieck-Teichmüller theory, as already noticed. In particular and in sharp constrast to what happens in the discrete case, the injective map of the proposition is very far from being an isomorphism. The next section ( §7) will be devoted to determining the first group, namely Aut( ČP (S)).

As a next step and thanks to the reconstruction theorem, more accurately Corollary 5.14, we find that the automorphism groups of Č(S), or equivalently of its 1-skeleton Č(S) (1) and that of Č * (S) coincide for d(S) > 1. (The cases d(S) = 0, 1 are well-known; besides the two 0-dimensional complexes occurring for d(S) = 1 are isomorphic: Č(S 0,4 ) Č(S 1,1 )). We record this piece of information as:

Proposition 6.3. For d(S) > 1, Aut( Č(S)) Aut( Č * (S)).
Our next result will require substantially more work. Recall that the type of a proloop, that is an element of Ľ(S) = Č(S) (0) , is well-defined, and more generally so is the type of any simplex σ ∈ Č(S). An automorphism φ ∈ Aut( Č(S)) is type preserving if it maps every simplex to one of the same type. In other words φ is type preserving if it preserves the Γ(S)-orbits: φ(σ) ∈ Γ(S) • σ for every σ ∈ Č(S). Before stating our next result, we remark that it does not use any notion of topology, dealing in principle with automorphisms which respect the simplicial structure, not necessarily the profinite topology. However the notion of type itself does require more structure; in fact it has not even been defined in the full profinite setting, and for good reasons. So we keep the notation Aut( Č(S)), denoting continuous simplicial automorphisms of Č(S), although some statements do not require continuity. We now state: Theorem 6.4. Let S be a connected hyperbolic surface; if S is not of type (1, 2), every simplicial automorphism of Č(S) is type preserving. If S = S 1,2 , an element of Aut( Č(S)) is type preserving if and only if it preserves the set of separating curves.

Note that here we assumed S to be connected for simplicity only. The statement for arbitrary hyperbolic surfaces is only slightly more involved and the extension is obvious. Moreover the statement is empty for d(S) = 0, 1 and these cases have been included only formally. From now on we assume that d(S) > 1. Then for d(S) = 2, either S = S 0,5 or S = S 1,2 , with the exceptional isomorphism C(S 0,5 ) C(S 1,2 ) and ditto for the respective congruence completions. This gives rise to the exception recorded in the statement.

We will break the bulk of the proof into two lemmas and then complete the proof of the theorem. First, except for type [START_REF] Anderson | Exactness properties of profinite completion functors[END_REF][START_REF] Artin | Etale Homotopy[END_REF], simplicial automorphisms preserve the set of separating curves. Lemma 6.5. Let S be connected hyperbolic, d(S) > 1, S not of type [START_REF] Anderson | Exactness properties of profinite completion functors[END_REF][START_REF] Artin | Etale Homotopy[END_REF]. Every simplicial automorphism φ ∈ Aut( Č(S)) maps a separating curve α ∈ Ľ(S) to a separating curve φ( α).

Proof. Suppose that α were nonseparating whereas φ(α) is separating. From Lemma 5.2 φ(α) must be of boundary type. There exist g, h ∈ Γ(S) such that both α = g • α and ϕ(α) = h • φ(α) are discrete curves. Moreover ϕ = h • φ • g -1 is also an automorphism of Č(S). Then the links of the vertices α and ϕ(α) in Č(S), namely Č(S α ) and Č(S ϕ(α) ), should be isomorphic. From our assumptions S α is of type (g -1, n + 2) while S ϕ(α) is of type (g, n -1). Then Theorem 5.1 implies that (g, n) ∈ {(1, 2), (1, 3)}.

In order to get rid of the case (g, n) = (1, 3) we closely follow the proof of ( [START_REF] Luo | Automorphisms of the complex of curves[END_REF], Lemma 2.2). Extend α to a pants decomposition {α, β, γ}, where β and γ are non-separating. Then (ϕ(α), ϕ(β), ϕ(γ)) is a 2-simplex of Č(S) and hence there exists k ∈ Γ(S) such that k • ϕ(α) = ϕ(α) and k • ϕ(β), k • ϕ(γ) are discrete curves which form a pants decomposition of S. Now ϕ(α) bounds a subsurface S 1,2 . Then, k • ϕ(β) and k • ϕ(γ) are contained in the subsurface S 1,2 and hence one of them, say k • ϕ(β), must be non-separating. Choose a simple curve which must be of the form ϕ( δ) in S 1,2 disjoint from k • ϕ(β); it bounds a subsurface S 1,1 of S 1,2 . Then ϕ( δ) ⊂ S 1,3 is a separating curve not of boundary type. By the proof of Lemma 5.5, δ is a proloop which is separating and not of boundary type on S 1,3 . On the other hand (ϕ(α), ϕ( δ), ϕ(β)) is a 2-simplex of Č(S) and hence (α, δ, β) is also a 2-simplex. There exists m ∈ Γ(S) such that m • α = α, m • β = β and m • δ = δ are discrete curves on S. Moreover, α and β are non-separating, δ is separating not of boundary type, while α, β and δ are pairwise disjoint. This is impossible and the claim follows.

Next it turns out that the requirement in the statement of the theorem concerning the exceptional case S = S 1,2 is actually general. In this lemma we will deal with curves (loops), that is elements of Ľ(S). It will then be easy to generalize this to multicurves i.e. arbitrary simplices of Č(S). So for the moment we state: Lemma 6.6. For any hyperbolic surface S, a simplicial automorphism which preserves the sets of separating classes of curves also preserves the type of the curves.

Proof. We may and do assume that d(S) > 1. The proof follows the lines of ( [START_REF] Luo | Automorphisms of the complex of curves[END_REF], Lemma 2.3). Let φ ∈ Aut( Č(S)) mapping separating elements of Ľ(S) to such. By lemma 5.2, φ preserves the set of proloops of boundary type. Let then α ∈ Ľ(S) be separating not of boundary type. As in the previous lemma, after left and right composition of φ with elements of Γ(S) we can assume that both α = α and φ(α) are discrete curves. The slit surfaces have two connected components:

S α = S 1 α ∪ S 2 α and S φ(α) = S 1 φ(α) ∪ S 2 φ(α)
, none of them of type (0, 3). Now, φ induces an isomorphism between the dual links φ : L - Č(S) (α) → L - Č(S) (α). The proof of Lemma 5.2 shows that L - Č(S) (α) is not chain connected and in fact it has exactly two connected components. A connected component L -,j Č(S) (α) of L - Č(S) (α) consists of those vertices of L - Č(S) (α) corresponding to the proloops γ on one connected component S j α of S α . As φ preserves chain connectedness, it must send a connected component of L -,j Č(S) (α) isomorphically to a connected component of L -,j Č(S) (φ(α)). Observe now that L -,j Č(S) (φ(α)) is the dual of a profinite curve graph, as it has the same set of vertices as Č(1) (S j α ) while two vertices are adjacent in L -,j Č(S) (φ(α)) if and only if they are not adjacent in Č(1) (S j α ). In particular φ induces isomorphisms Č(1) (S j α ) → Č(1) (S j φ(α) ). These are flag complexes by Proposition 6.1 and we get isomorphisms Č(S j α ) → Č(S j φ(α) ). We can use now Theorem 5.1 to derive that either α and φ(α) have the same topological type or else:

(1) S 1 α = S 2 φ(α) = S 1,1 , S 2 α = S 1 φ(α) = S 0,4 ; (2) S 1 α = S 2 φ(α) = S 1,2 , S 2 α = S 1 φ(α) = S 0,5 ; (3) S 1 α = S 1,1 , S 1 φ(α) = S 0,4 , S 1 α = S 0,5 , S 2 φ(α) = S 1,2 .
None of these cases can occur since an isomorphism Č(S 1 α ) → Č(S 1 φ(α) ) will necessarily send a nonseparating curve β to a separating one of S 1 φ(α) , hence of S φ(α) . This would contradict Lemma 6.5. The claim follows.

End of proof of Theorem 6.4. We use induction on the dimension of the simplex σ ∈ Č(S). Lemma 6.6 yields the claim when the dimension 0. Assume it holds true up to dimension k -1 and let σ be a kdimensional simplex. After composing φ on the left and on the right φ by two elements of Č(S) we may assume that σ = (α 0 , α 1 , . . . , α k ), φ(σ) = (β 0 , β 1 , . . . , β k ) are both discrete simplexes. By the induction hypothesis σ = (α 0 , α 1 , . . . , α k-1 ) and φ(σ ) = (β 0 , β 1 , . . . , β k-1 ) lie in the same Γ(S)-orbit. Indeed by Proposition 3.7, they are in the same Γ(S)-orbit, namely they have the same topological type. Composing further with an element of Γ(S) we may assume that σ is fixed pointwise: φ(α i ) = α i , i = (0, 1, . . . , k -1). Denote by the same letters the traces of the curves α k and β k on the surface S σ hyperbolic of dimension d(S)-k but not necessarily connected. However Lemma 6.6 still holds true in that case (for obvious reasons), hence α k and β k have the same topological type on S σ . An element of Γ(S σ ) sending α k to β k lifts to a mapping class in Γ σ (S) ⊂ Γ(S) which maps σ to φ(σ), proving the claim and completing the proof of Theorem 6.4. 6.2. Automorphisms of goups and complexes. In this short subsection we make the connection between group automorphisms on the one hand, automorphisms of complexes on the other. The next statement will serve to emphasize how computing the automorphisms of curve complexes enables one to study, not only the automorphism groups Aut( Γ(S)) of the procongruence modular groups, but indeed the groups Aut( Γλ (S)) for all values of λ ∈ Λ, that is the automorphism groups of the open subgroups of the procongruence modular groups. In the discrete setting the analogous statement comes from Theorem 2.4 and leads, via Theoreme 2.1 to the statement of Corollary 2.5. In the procongruence (or profinite) setting, we first define inertia preserving automorphisms just as in the discrete case, namely: Definition 6.7. An element of Aut( Γ(S)) is inertia preserving if it globally preserves the set of procyclic subgroups generated by Dehn twists, that is maps a twist in Γ(S) to a profinite power of a twist.

We denote again with an upperscript the subgroup Aut * ( Γλ (S)) ⊂ Aut( Γλ (S)) of the inertia preserving automorphisms. Here however the analog of Theorem 2.4, asserting that every automorphism preserves inertia, although conjectured to hold true, is not available. We only remark that this statement stands in close analogy with the so-called local correspondence of anabelian geometry. So we will deal explicitly with the subgroup of inertia preserving automorphisms and we do indeed restrict attention to automorphisms, as opposed to the more general isomorphisms appearing in Corollary 2.5. This is purely for the sake of simplicity. The extension to isomorphisms would be easily available. In this context we have: Proposition 6.8. For every hyperbolic S and every congruence level λ ∈ Λ there is a natural morphism:

γ λ : Aut * ( Γλ (S)) → Aut( Č(S)).
This morphism is injective if Γ λ (S) has trivial center, thus in particular if Γ(S) itself has trivial center.

Proof. Note that in the last assertion we refer to the centers of the discrete groups, which actually coincide with those of the completed ones (cf. [START_REF] Boggi | On the procongruence completion of the Teichmüller modular groups[END_REF], Corollary 6.2). Since we are working with colored modular groups, the only exceptions are the types (1, 1) and (2, 0), in which cases the center is of order 2, generated by the hyperelliptic involution. One should also pay attention to the levels such that Γ λ (S) ⊂ Γ(S) contains this involution. These cases could easily be treated in detail but we refrain to do so here.

The main remark and the main point in this proof consists in the fact that given φ ∈ Aut * ( Γλ (S)) one can assign to every simplex α ∈ Č(S) an image in a coherent way, thereby defining γ λ (φ) ∈ Aut( Č(S)). This is a direct consequence of Proposition 4.

φ(U λ σ ) = U φ(σ)
, which is valid for every simplex σ ∈ Č(S) and every congruence level λ ∈ Λ. One should pay attention to the exact meaning of this formula. Indeed on the right-hand side U φ(σ) denotes a kind of "generic"' open subgroup of the group G φ(σ) . It is asserted, in accordance with the above, that there exists a unique simplex φ(σ) ∈ Č(S) such that the left-hand side, namely φ(U λ σ ), is open in G φ(σ) ; this property defines φ = γ λ (φ). We have thus defined a map γ λ for every λ ∈ Λ. It is actually easy to see that this a coherent family with respect to the level λ. More precisely consider µ ≥ λ, so that Γ µ (S) ⊂ Γ λ (S) and assume that Γµ (S) is invariant (characteristic) in Γµ (S) (recall that these groups are topologically finitely generated, so that invariant subgroups are cofinal). Then there is a natural restriction map ρ λ,µ : Aut * ( Γλ ) → Aut * ( Γµ ) and it is clear that γ λ = γ µ • ρ λ,µ .

We finally address the issue of the injectivity of the map γ λ . We use the natural action of Γ(S) on Č(S), which defines an injective map Inn( Γλ (S)) → Aut( Č(S)). Moreover, for every λ ∈ Λ, φ ∈ Aut * ( Γλ (S)), g ∈ Γλ (S) and σ ∈ Č(S), we find that:

φ(g)(σ) = φ • g • φ-1 (σ) (with φ = γ λ (φ)). If Γ λ (S)
is centerfree, so is Γλ (S) as mentioned above i.e. Γλ (S) = Inn( Γλ (S))). Then if φ = id the formula above implies that φ(g)g -1 ∈ Z( Γλ (S)) hence φ(g) = g for all g ∈ Γλ (S); in other words φ = id, proving injectivity and completing the proof. 6.3. The arithmetic Galois action. We remark now that the above makes it possible to define a faithful arithmetic Galois action on the completed curve complex. We will stick here to the basic and most important case, namely the action of G Q , the absolute Galois group of the field Q, on the curve complex. Recall that for S hyperbolic connected, the (Deligne-Mumford) moduli stack M(S) is defined over Q, hence a natural outer action G Q → Out( Γ(S)). Here the full profinite completion Γ(S) actually stands for the geometric étale fondamental group: Γ(S) = π 1 (M(S) ⊗ Q). Very little is known about this action but two pieces of information are quite relevant here. First it is inertia preserving, as initially showed by A.Grothendieck and J.Murre (see [START_REF] Lochak | Groupe fondamental des champs algébriques, inertie et action galoisienne[END_REF] for references and much more on this and related topics); second it is faithful for d(S) > 0 as a consequence of Belyi's theorem. It is easy to see that this action descends to the congruence quotient Γ(S) and remains faithful, because in particular Γ(S) = Γ(S) for d(S) ≤ 5. Moreover the outer action can be (non canonically) lifted to a bona fide action by picking a (possibly tangential) rational basepoint on the moduli stack M(S). All in all, after picking a rational basepoint we get a faithful inertia preserving action G Q → Aut * ( Γ(S)) for d(S) > 0 (we again refer to [START_REF] Lochak | Groupe fondamental des champs algébriques, inertie et action galoisienne[END_REF] for much more background, references, etc.). By composing with the map γ ol Proposition 6.8 (γ = γ λ for λ the trivial level: Γ λ (S) = Γ(S)) we get: Proposition 6.9. Let S be connected hyperbolic with d(S) > 0; then there is a map:

G Q → Aut * ( Č(S))
which is injective and canonical up to composition with the action of Inn( Γ(S)) on Č(S).

For d(S) > 1 the same holds true for the graph Č * (S).

Proof. The possible composition by an inner automorphism of Γ(S) comes from the choice of a rational basepoint. In other words the proposition asserts the existence of a natural faithful outer action of G Q on Č(S). Here the only thing which requires proof is the injectivity in the two cases (types (1, 1) and (2, 0)) where Γ(S) has nontrivial center. But the kernel of the map γ of Proposition 6.8 is then generated by an involution. Now any involution in G Q is conjugate to complex conjugacy so that it is enough to check that the image of this latter element is not central; but this is clear since it corresponds to a reflection of the surface. We thus find that the image of G Q in Aut( Γ(S)) does not intersect the kernel of γ, which completes the proof for the curve complex. The last assertion comes either from the reconstruction theorem or from the fact that, for d(S) > 1, Č * (S) identifies with the 1-skeleton of the dual of Č(S).

It should be stressed that we get a faithful action of the arithmetic Galois group on a profinite space, whereas it is more common to get an action on a profinite group, which itself arises as a cohomological or homotopical invariant of an underlying "classical" space. Actually, given -say -a geometrically connected scheme defined over Q, one can make its étale covers into a p(r)oset by considering a (pro)point in the (pro)universal cover, then let G Q act on this proset, much as is done with "dessins d'enfants". These correspond to the type (0, 4) and simply give a "pictionary" of the finite (étale, that is here simply unramified) covers of M 0,4 (C), alias P 1 (C) \ {0, 1, ∞} alias C \ {0, 1}; recall also the isomorphism C(S 0,4 ) C(S 1,1 ) and that these complexes are 0-dimensional. The resulting Galois action is indeed faithful as an easy corollary of Belyi's theorem, but it is essentially no easier to study than the usual action on the geometric fundamental group. Here curve complexes retain a kind of homotopical information at infinity from the tower of covers of the moduli stacks in all dimensions and are much more amenable to a direct study.

The above Galois action readily extends to an action of the Grothendieck-Teichmüller group IΓ as defined in [START_REF] Hatcher | On the Teichmüller tower of mapping class groups[END_REF] ( see also [START_REF] Nakamura | On a subgroup of the Grothendieck-Teichmüller group acting on the tower of profinite Teichmüller modular groups[END_REF]) for background material) essentially by the very definition of this group which however we skip here, as it would lead us too far afield. For the sake of clarity, we record this explicitly as Proposition 6.10. Let S be connected hyperbolic with d(S) > 0; then there is a map:

IΓ → Aut * ( Č(S))
which is canonical up to composition with the action of Inn( Γ(S)) on Č(S) and is injective if Γ(S) is centerfree. For d(S) > 1 the same holds true for the graph Č * (S).

Note that here we cannot a priori exclude the existence of a nontrivial kernel in the two cases when Γ(S) has nontrivial center. It may be useful to remind the reader that there is a nested sequence of profinite groups:

G Q ⊂ IΓ ⊂ GT ⊂ Aut * ( F2 ),
where GT is the original "genus 0" Grothendieck-Teichmüller group introduced by V.Drinfeld, IΓ is the version adapted to all genera constructed in [START_REF] Hatcher | On the Teichmüller tower of mapping class groups[END_REF] and [START_REF] Nakamura | On a subgroup of the Grothendieck-Teichmüller group acting on the tower of profinite Teichmüller modular groups[END_REF], whereas F 2 = Z * Z denotes the free group on 2 generators. For any S as in the proposition, there is also an injective map IΓ → Aut * ( Γ(S)) giving rise to a canonical injection IΓ → Out * ( Γ(S)). If S has genus 0 we can enlarge IΓ to GT both here and in Proposition 6.10, that is both in the group and complex theoretic frameworks. Finally it is essential that both Propositions 6.9 and 6.10 are "badly" wrong for the pants complex ČP (S), as will become clear in the next section. We also refer the reader to [START_REF] Lochak | Automorphism groups of profinite complexes of curves and the Grothendieck-Teichmüller group[END_REF][START_REF] Lochak | Results and conjectures in profinite Teichmüller theory[END_REF] for much more on these and related topics.

7. Rigidity of the procongruence pants complex 7.1. Main result. We now turn to the study of the automorphism group of the procongruence pants complex ČP (S), where S is hyperbolic, changing gear in essentially two ways. First in this section we will have to use topological stacks rather than CW-complexes, as will be detailed in the next paragraph. Second, C P (S) will denote the full two-dimensional pants complex and not the pants graph, namely the 1-skeleton C

P (S). Let us state right away that we will use two specific ingredients in the proof of the rigidity result (Theorem 7.1 below): first the rigidity of the discrete pants complex C P (S) ([24] or Theorem 2.13 above), second the simple connectedness of that same complex ( [START_REF] Hatcher | On the Teichmüller tower of mapping class groups[END_REF],Theorem D in the introduction there)

It is worth insisting on this second crucial ingredient because although we will have to introduce somewhat sophisticated objects to make sense of the result, the simple connectedness of C P (S) remains a central tenet of the proof. Indeed that result features an incarnation of the central fact in the foundation of Grothendieck-Teichmüller theory. It is central in [START_REF] Nakamura | On a subgroup of the Grothendieck-Teichmüller group acting on the tower of profinite Teichmüller modular groups[END_REF] (a sequel and improvement of [START_REF] Hatcher | On the Teichmüller tower of mapping class groups[END_REF]), and it is tightly connected with the so-called "two-level principle" which itself can be geometrically translated into the fact that for d(S) > 2 the fundamental group of the moduli stack M(S) is "concentrated at infinity", as stated by A. Grothendieck in his Esquisse and vindicated in [START_REF] Lochak | The fundamental group at infinity of the moduli spaces of curves[END_REF] (see also [START_REF] Boggi | Profinite Teichmüller theory[END_REF]). It is thus rather interesting that one of the main foundational results of the present paper, namely the rigidity of the procongruence (or in fact profinite -see below) pants complex, can be considered a direct corollary of the main foundational result of the theory.

Let us now state the result rather bluntly. We will return to a more careful elucidation of the definitions in the next paragraphs ( § §7.2, 7.3). Actually the results in this section hardly depend on the type of completion, provided it is fine enough, in particular residually finite. So let us denote by a prime ( ) a completion which sits between the procongruence and the full profinite one. In other words we pick an inverse system of levels (cofinite subgroups of Γ = Γ(S)) which contains the congruence system Λ and of course is contained in the full system M (see §A.10). The reader who is willing to make life simpler or lighter is welcome to elect Λ and stick to the congruence completion. Indeed for ease of notation, below we will refer to our fixed inverse system as Λ.

There are natural epimorphisms: Γ Γ Γ, and ditto for the other completed objects. The group Γ is residually finite (i.e. there is a natural embedding Γ → Γ ) since Γ is. Of course if the congruence conjecture holds true (which we do not assume here) all three completions coincide. We will also be interested in the respective centers of these profinite group. This is known only for the conguence completion (see [START_REF] Boggi | On the procongruence completion of the Teichmüller modular groups[END_REF]) : one has Z( Γ) = Z(Γ) so that Inn( Γ) = (Inn(Γ)) ∨ .

As mentioned above we now have to consider the two-dimensional procomplex C P (S) as a topological prostack, which we denote simply C P (S) st . Precise definitions are given in §7.2 below, along with the exact meaning of the automorphism group. Granted these for the moment we can state: Theorem 7.1. For every S connected hyperbolic and any completion which is finer than the congruence one, the group Aut(C P (S)) st of automorphisms of the topological prostack C P (S) st is determined by the following split short exact sequence:

1 → (Inn(Γ(S)) → Aut(C P (S) st ) → Z/2 → 1.
In other words and more concisely :

Aut(C P (S) st ) = Aut(C P (S)) .
Note that we get (Inn(Γ(S)) on the left-hand side of the short sequence of the theorem, rather than Inn(Γ(S) ) . As mentioned above these groups coincide in the procongruence case.

So Out(C P (S) st ) Z/2, like in the discrete case, and the sequence is split by complex conjugacy. Let us briefly detail the one-dimensional case, noting that in dimension 0 (S = S 0,3 ), all complexes are empty and there is nothing to prove. If d(S) = 1, the type is (0, 4) or (1, 1), the congruence conjecture holds true and we are dealing with ĈP (S 1,1 ) st ĈP (S 0,4 ) st Fst , the profinite Farey tesselation viewed as a pro-object in the category of one-dimensional DM-stacks (orbifold curves). We will review this exceptional case in §7.5.

Let us briefly elucidate the two and three dimensional cases, where the congruence conjecture has been vindicated. In dimension [START_REF] Boggi | Galois coverings of moduli spaces of curves and loci of curves with symmetry[END_REF] of index 5 and self-normalizing (see §A.4). This takes care of the left-hand side of the exact sequence in dimension 2. Recall that the pants graphs for types (0, 5) and (1, 2) are not isomorphic (cf. Prop. 5.15). Finally in dimension 3, the congruence conjecture is still valid, the Teichmüller groups Γ 0,6 and Γ 1,3 have trivial centers and we just record the fact that Z(Γ 2 )

2, Γ 1,[2] = Γ 1,2 × Z (direct product) and Γ 1,[2] /Z = Γ 1,2 ⊂ Γ 0,
Z/2 (generated by the hyperelliptic involution) with Γ 2 /Z Γ 0, [START_REF] Boggi | Characterizing closed curves on Riemann surfaces via homology groups of coverings[END_REF] . For all the other types (g, n) the centers of the discrete and procongruence groups are trivial:

Inn(Γ(S g,n )) ∨ = Inn( Γ(S g,n )) = Γg,[n] .
We stated the result above in terms of topological prostacks, which may appear as somewhat fancy sounding objects. Let us then try to informally answer the natural question: why are these needed ? First the main object of study is the projective system of quotients (C P (S)/Γ λ ) λ∈Λ , which can be viewed as a pro-object of some category. So it is only natural that pro-categories pop out; we review the basic inputs in §7.3 below.

Now in what category should we consider the quotient C P (S)/Γ λ , assuming that λ is large enough and possibly (w.l.o.g.) that Γ λ is normal in Γ? At some point we will want to lift morphisms or automorphisms of these quotients to C P (S). The latter is indeed simply connected, but not of course the pants graph C

(1) P (S). So we need to use the full two-dimensional complex in the proof. Moreover lifting a morphism usually presupposes that we are dealing with an étale (say unramified; flatness does not enter here) cover. So the projection C P (S) → C P (S)/Γ λ should be unramified, which it is if we consider the stack quotient. However this case is not covered by the usual (algebraic) theory of stacks (see e.g. [START_REF] Laumon | Champs algébriques[END_REF]), nor by Thurston's theory of orbifolds, an important point being that the action of Γ λ on C P (S) is not proper and discontinuous. (Of course we could consider things "bottom up", C P (S)/Γ λ being a cover of C P (S)/Γ with finite group Γ/Γ λ , but it doesn't help.) Already stabilizers are not finite. We will see below that B. Noohi's theory of topological stacks is in fact taylored to our needs.

A last remark: the pants graph affords a geometric and in fact modular interpretation, first noted in some form by D. Mumford and briefly recalled in 7.6 below. So it is immediately connected with complex theoretic, hyperbolic, algebraic and in fact arithmetic structures. However, to the best of our knowledge, this is not the case of the pants complex itself. In other words we do not have a complex theoretic interpretation of the elementary homotopies which make up the two-cells of the pants complex. This is another deep reason why we have to work with topological stacks. 7.2. Topological stacks. The foundations of the theory of topological stacks were developed by B. Noohi in [START_REF] Noohi | Foundations of Topological Stacks[END_REF], unfortunately unpublished but easily available. We will quote extensively from that paper, which we recommend for further reading, along with later papers of that author, especially [START_REF] Noohi | Fundamental groups of topological stacks with the slice property[END_REF]. For the theory of the fundamental group of algebraic stacks, see [START_REF] Noohi | Fundamental groups of algebraic stacks[END_REF][START_REF] Lochak | Groupe fondamental des champs algébriques, inertie et action galoisienne[END_REF]. However, in truth we will use very little of the theory and our situation is a particularly simple and favorable one. So we will try and avoid doing too much overshooting, confining ourselves to some pointed reminders and observations.

Start from the category Top of (compactly generated) topological spaces, our main example here being X = C P (X), a two-dimensional non locally finite CW-complex. Recall that in this case the topology is the quotient topology associated to the gluing of the cells. We cannot go over rather long definitions (referring globally to [START_REF] Noohi | Foundations of Topological Stacks[END_REF]) but to be precise, we mention that defining the category TopSt of topological stacks over Top involves the choice of a family of local fibrations ; here we work with homeomorphisms, which is item 6 in [START_REF] Noohi | Foundations of Topological Stacks[END_REF], Example 13.1. Now we consider the action groupoid [Γ λ × X ⇒ X] ([27], §7). This is a topological groupoid and it is easily seen that the action of Γ λ is mild at every point. Here we should insist that this does not entail that the stabilizer groups are finite; they are not. In fact consider a vertex of X, which is nothing but a maximal multicurve (a.k.a. "pants decomposition") of X. Then its stabilizer under the action of the full Teichmüller group Γ = Γ(S) is an extension of a finite symmetry group by the free abelian group of rank d = d(S) generated by the twists along the curves which span the given multicurve. For λ large enough the stabilizer is thus given as a cofinite subgroup of the latter group, that is a lattice in Z d (with finite quotient).

Thus we find that the quotient of the groupoid [Γ λ × X ⇒ X] is a topological Deligne-Mumford stack in the sense of [START_REF] Noohi | Foundations of Topological Stacks[END_REF], Def. 14.3. Here again the adjective "topological" is essential and does not only refer to the fact that all the operations are compatible with a given topology. It implies an effective enlargement of the category of admissible stacks. In fact every C-stack (locally of finite type) can be regarded as a topological stack ( [START_REF] Noohi | Foundations of Topological Stacks[END_REF], §20, plus a GAGA-type result). On the other hand, the usual orbifolds are also included in the picture ( [START_REF] Noohi | Foundations of Topological Stacks[END_REF], §19.3) as a full subcategory of topological Deligne-Mumford stacks. Note that all these categories are in fact 2-categories but for the sake of simplicity we will not record this, just as we do not explicitly state that assertions are often up to 2-morphisms.

We denote the quotient by X λ or by (C P (X)/Γ λ ) st . We are indeed in a comparatively simple situation, Deligne-Mumford stacks being in general of this form only locally (this is also true in the algebraic setting, for the étale topology; see [START_REF] Laumon | Champs algébriques[END_REF] or [START_REF] Lochak | Groupe fondamental des champs algébriques, inertie et action galoisienne[END_REF]). Here, since X, regarded as element of Top or TopSt is simply connected, it is in fact the universal cover of X λ (see also [START_REF] Noohi | Foundations of Topological Stacks[END_REF], Theorem 18.24). Moreover, the naive quotient C P (X)/Γ λ in Top is the coarse moduli space of X λ . We therefore denote it X λ mod ; there is a functorial morphism X λ → X λ mod from the topological stack X λ to its coarse moduli space. Now the proof of Theorem 7.1 will be quite short (see below), based, apart from the two main and specific ingredients mentioned above, on a few inputs from the covering theory for topological stacks ( [START_REF] Noohi | Foundations of Topological Stacks[END_REF], §18) and some notions which are common to topological and algebraic stacks. We refer to [START_REF] Noohi | Foundations of Topological Stacks[END_REF], §3, for a quick but sufficient review, or e.g. to [START_REF] Laumon | Champs algébriques[END_REF]. 7.3. Pro-objects and their automorphisms. In this short paragraph we recall how pro-objects are handled. In view of the relative concreteness of our situation we need only a few basic inputs, to be found essentially in the Appendix to [START_REF] Artin | Etale Homotopy[END_REF]. For more and a more modern approach in the framework of model categories, see [START_REF] Quick | Group action on profinite spaces[END_REF][START_REF] Quick | Some remarks on profinite completion of spaces[END_REF] which are quite relevant in the framework of this paper.

First recall some vocabulary: an inverse or projective limit (lim ←-) is just a limit, an inductive one (lim -→ ) is a colimit, an inverse or projective system is a particular case of a cofiltering category. We use a generic Λ, which in our case concretely denotes any inverse system which is finer than the one defining the congruence completion. To make the connection with categories, simply declare that there is a (unique) morphism λ → µ if and only if λ ≥ µ. Now indeed for every pair λ, µ of elements of Λ there exists ν ∈ Λ such that ν → λ and ν → µ. Next we consider pro-objects (rather than just limits) associated with a small category C, here especially the category tStacks of topological stacks. These are given as coherent collections X = (X λ ) λ∈Λ , equivalently as maps Λ → X, with X λ ∈ C for all λ ∈ C (or in fact just for λ large enough).

These are made into a category pro -C by defining morphisms between two pro-objects X and Y :

Hom(X, Y ) = colim µ lim λ Hom(X λ , Y µ ).
Here beware of the fact that we are considering a cofiltering category Λ as the primary object, rather than the opposite filtering category Λ o . So we get a contravariant functor as usual but limits and colimits are swapped. In any case this essentially amounts to describing the respective variances of "source" and "image" by chains of morphisms λ → λ → µ → µ (i.e. λ ≥ λ ≥ µ ≥ µ in the case of inverse systems). Although we do not seriously need it, a useful result says that by a clever reindexing we can bring a morphism of pro-objects to a more manageable form, namely Proposition 7.2. Let f : X → Y be a map in pro -C for a small category C, with indexing cofiltering category Λ ; then it can be represented, up to isomorphism, by an inverse system of maps (φ λ : X λ → Y λ ) λ∈Λ . Moreover f is invertible if and only if there exists a system (ψ λ :

X λ → Y λ ) λ∈Λ with ψ λ • φ λ = id.
Proof. This is essentially Corollary 3.2 in the Appendix of [START_REF] Artin | Etale Homotopy[END_REF]. This statement is an especially useful but rather particular case of Proposition 3.3 (ibidem; see also [SGA 4]). Note that the system of maps (φ λ ) λ∈Λ can be viewed as a pro-object in the category of maps in C, which helps unraveling the definition of pro -C, the procategory built from C. The addition on invertibility in the statement comes as a particular case of Scholie 3.5, loc. cit..

We should warn the reader that the word "represent" in the statement is pointing to the fact that this is "up to an equivalence" i.e. one may have to replace X and Y by equivalent, that is isomorphic pro-objects, isomorphisms being invertible morphisms. In particular this proposition is dealing with a given morphism f , not with the whole set Hom(X, Y ).

To sum up, we will be working in the category pro-TopSt of topological prostacks, more particularly with C P (S) st , defined by the coherent sequence (X λ ) λ∈Λ where X λ = (C P (S)/Γ λ ) st . Recall that the type of the completion ( ) is defined by the choice of the inverse system (cofiltering category) Λ. We sometimes abbreviate C P (S) st to X st , keeping the bare X for the discrete complex C P (S), viewed as a topological space (CW-complex), a fortiori a topological stack. An automorphism φ ∈ Aut(C P (S) st ) = Aut(X st ) can be defined by two systems of maps (φ λ , ψ λ : X λ → X λ ) for λ ∈ Λ large enough, which are isomorphisms (equivalences) of the stacks X λ = (C P (S)/Γ λ ) st and are inverse of each other (ψ λ • φ λ = φ λ • ψ λ = id).

7.4. Proof of Theorem 7.1. After these preliminaries we may now return to our situation ; the proof of Theorem 7.1 is now quite short. Start with φ ∈ Aut(C P (S) st ) = Aut(X st ) an automorphism of the topological prostack C P (S) st , given by a coherent sequence of morphisms of stacks φ λ : X λ → X λ with X λ = (C P (S)/Γ λ ) st . For λ ∈ Λ large enough (i.e. Γ λ small enough), φ λ is an isomorphism, in particular it is a representable morphism.

Fixing λ, let then p λ : X → X λ denote the canonical projection (X = C P (S)). The map p λ is a covering map of topological stacks (see [START_REF] Noohi | Foundations of Topological Stacks[END_REF], Def. 18.10) and X is simply connected. We can thus apply the general lifting lemma (see [START_REF] Noohi | Foundations of Topological Stacks[END_REF], Prop. 18.18) to p λ and the map φ λ • p λ . It yields a map φλ : X → X, which again is invertible for λ large enough. So φλ is nothing but an automorphism of the (discrete) pants complex C P (S). We now appeal to Margalit's rigidity result ([24] or Theorem 2.13 above) asserting that such an automorphism is induced by an element of M od(S). By varying λ ∈ Λ along the projective system defining the completion prime ( ), we get the assertion of Theorem 7.1.

Let us add two remarks. First the significance of this rigidity result lies in it being the seed of a profinite version of Grothendieck-Teichmüller theory, much in the spirit of the Esquisse. One should insist again that the completed curve complex, say Č(S) st , is far from rigid. Indeed its automorphism group appears as a version of the Grothendieck-Teichmüller group and thus contains the Galois group Gal(Q); it is independent of S for S "generic" enough. It is important to note the fact restricting ourselves to morphisms of stacks does not represent a serious restriction in this context. The point is that the Grothendieck-Teichmüller group clearly induces such morphisms, a fortiori so does the arithmetic Galois group. We hope to return to these facts in more detail elsewhere (see however [START_REF] Lochak | Automorphism groups of profinite complexes of curves and the Grothendieck-Teichmüller group[END_REF] for a draft version)..

The second remark is that we applied a slightly paradoxical strategy. There appear two specific ingredients in the proof of Theorem 7.1, namely the simple connectedness of the discrete pants complex C P (S) and its rigidity. In turn this last result is shown by D. Margalit (in [M]) by appealing to the rigidity of the discrete curve complex C(S) ( [START_REF] Ivanov | Automorphisms of complexes of curves and of Teichmüller spaces[END_REF][START_REF] Luo | Automorphisms of the complex of curves[END_REF] or Theorem 2.1 above). In other words we proved the rigidity of the completion of the pants complex exploiting the rigidity of the discrete curve complex, whose completion in turn is anything but rigid. 7.5. The one-dimensional case: a review. In this subsection we detail the geometry of the onedimensional case, making the connection with "dessins d'enfants" and reproving Theorem 7.1 in this particular case (see Proposition 7.3 below) in a simpler way. The perceptive reader will find that we could actually improve the statement in this one-dimensional case, but for simplicity we refrain from doing so. For the higher dimensional situation, see §7.6 below. In dimension 1 the modular groups read Γ 0, [START_REF] Boggi | On the procongruence completion of the Teichmüller modular groups[END_REF] Γ 1,1 /Z P SL 2 (Z) (Z Z/2) where we use the quotient of Γ 0, [START_REF] Boggi | On the procongruence completion of the Teichmüller modular groups[END_REF] which acts effectively. Since C P (S 0,4 ) = C P (S 1,1 ) = F we may and will restrict attention to S = S 0,4 . Then M 0,4 ∼ = P 1 \ {0, 1, ∞} which for ease of notation we denote M in this paragraph. So up to a stacky phenomenon which needs not concern us here (see [START_REF] Lochak | Groupe fondamental des champs algébriques, inertie et action galoisienne[END_REF] for much more detail) the (rigidified) version of M 0, [START_REF] Boggi | On the procongruence completion of the Teichmüller modular groups[END_REF] is given by the quotient M 0,4 /S 3 . Here S 4 permutes the 4 punctures and acts effectively via S 4 /V S 3 , where V is the Klein 4-group. This is summarized by the short exact sequence 1 → Γ 0,4 → Γ 0, [START_REF] Boggi | On the procongruence completion of the Teichmüller modular groups[END_REF] → S 3 → 1.

Recalling that the congruence conjecture holds true in this case, Theorem 7.1 translates into: Here Fst denotes the Farey tesselation viewed as a (pro-)orbifold and the right-hand factor in the attending short exact sequence (see the statement of Theorem 7.1) is generated by the class of the matrix diag(-1, 1).

For a bit of geometry, we start again with M ∼ = P 1 \ {0, 1, ∞}. Then M P 1 with boundary divisor ∂M = {0, 1, ∞}. We denote the analytic version also F = M(C) = P 1 (C) anticipating on the higher dimensional cases to be tackled below. Next F = P 1 (C) is naturally triangulated into two triangles, say black and white, by the two closed hemispheres, with vertices {0, 1, ∞}. The common boundary is the equator, i.e. the set M(R) of the real points inside M(C) or more algebraically the set of fixed points of the complex conjugacy, generating the Galois group Gal(C/R). Now the attending Teichmüller space T 0,4 T 1,1 (see §A.2) is the Poincaré upper half-plane H or equivalently (i.e. up to a fractional transformation) the Klein disk D, equipped with the respective actions of P SL 2 (Z). Lifting the ideal triangulation of M via the natural projection T 0,4 = H → P 1 \ {0, 1, ∞} = M(= M 0,4 ) we get the classical bicolored Farey tesselation F of the upper half-plane H or the disk D. We also get the projective system of (F λ ) λ∈Λ defined by all the finite covers of M viewed as orbifold (a.k.a. DM C-stacks): F λ = F/Γ λ = (C P (S)/Γ λ ) st . Note that F λ determines a triangulation (tesselation) of the completed level M λ . The canonical projection π λ : M λ → M, is an unramified orbifold cover.

Let now φ ∈ Aut( Fst ) given by a compatible system of maps φ λ : F λ → F λ which are invertible for λ large enough. In a more algebraic language: M λ = X λ is an algebraic curve which is actually defined over Q (i.e. it is defined over some finite extension of Q). The projection π λ : X λ → X 0 = P 1 \ {0, 1, ∞} = M is an (algebraic) Belyi map (also defined over Q) determining an ideal triangulation of X λ . It extends to a map (still π λ ) M λ = X λ → X 0 = M between the respective completions. We denote the "ground level" with a zero, writing Γ 0 = Γ, M 0 = M, M 0 = M, etc. The projective algebraic curve X λ has quadratic singularities (nodes) lying over the points 0, 1 and ∞ for the projection π λ . Its normalization X λ is a smooth, not necessarily connected, projective algebraic curve, also defined over Q as well as every connected component. One can view the (dual of) the triangulation of X λ as a "dessin d'enfant" drawn on the curve viewed as a topological surface and rigidifying the situation entirely. For this translation we refer to [START_REF] Voevodskii | Equilateral triangulations of Riemann surfaces and curves over algebraic number fields[END_REF] and many other papers. Note that we are not using the astonishing part of Belyi's theorem, which asserts that every algebraic curve defined over Q arises in this way. For instance the fact that X λ is affine algebraic defined over Q was known to A.Weil, as well as everything that is mentioned above (see e.g. [START_REF] Weil | The field of definition of a variety[END_REF] which introduces the notion of "descent" and contains all the necessary material -and more). We may consider φ λ as a differentiable automorphism of M λ , up to isotopy. Every vertex, resp. edge, resp. face (triangle) of the tesselation is mapped to another such, which determines a diffeomorphism up to isotopy. Actually much more is true since the situation can be uniquely rigidified. Namely φ λ a priori permutes the connected components of the normalization X λ and on each of these components it is isotopic to a unique map which is either algebraic or "antialgebraic" i.e. algebraic after composing with the complex conjugacy (which is a well-defined involution), according to whether it preserves or inverts the orientation determined by the complex structure. One then arrives at the one-dimensional case of Proposition 7.6 below, which we refrain to state here. For the more topologically inclined reader we note that part (not all) of the above can be expressed in another language, namely that of the so-called flat surfaces (see e.g. [START_REF] Troyanov | On the moduli space of singular Euclidean surfaces[END_REF] and references therein). Now for every λ ∈ Λ there is a canonical projection p λ : H → M λ and H is simply connected ; indeed it is the universal cover of M λ for every λ ∈ Λ. Then φ λ lifts to φλ : H → H, preserving the Farey tesselation F of H. The lift φλ is well-defined up to the left action of Γ λ . This implies that φλ is (isotopic to) an element of P GL 2 (Z) ∼ = Aut(F ). It belongs to P SL 2 (Z) if and only if the original map φ λ is orientation preserving which is now seen to be a global property, that is independent of the choice of a connected component of X λ . We thus get a compatible system of elements φλ ∈ P GL 2 (Z), that is an element of P GL 2 (Z) ∼ = P SL 2 (Z) Z/2, which completes the proof of Prop. 7.3. The pattern followed above is the same as in the general case of Theorem 7.1, only simpler. Indeed H is the universal cover of M λ (recall that M = P 1 \ {0, 1, ∞}) for every λ ∈ Λ in the ordinary (topological) sense of the word and here C P (S 0,4 ) H because the 2-cells of the pants complex arise from triangles only (pentagons and faces of type 6AS do not exist in dimension 1). So this is the only case where the full pants complex is realized in a geometric, in fact modular way. 7.6. A geometric interlude: Fulton curves and the pants graph. We close this section with some geometric observations, underlining the modular significance of the pants graph. Unfortunately the geometric significance of the full two-dimensional complex C P (S) is not clear at the moment. Recall that in some sense its 2-cells come from elementary homotopies which generalize the classical Mac Lane's relations in genus 0 (i.e. in braided categories). This paragraph does not contain any really new result and we will be somewhat sketchy but we hope that such geometric observations can nonetheless prove suggestive and useful.

Starting from our usual connected hyperbolic surface S (d(S) ≥ 1) we consider again the attached moduli space M = M(S), viewed here as a complex orbifold (see §A.2), M the stable (Bers-Deligne-Mumford) compactification of M. The boundary divisor ∂M = M \ M classifies Riemann surfaces with nodes of the same type as S, or one-dimensional proper complex D-M stack with quadratic singularities and finite groups of automorphisms, in a more algebraic language. Passing to a level structure λ ∈ Λ, that is a representable étale cover M λ → M, we then get a compactification M λ with divisor at infinity ∂M λ = M λ \ M λ . We In other words F = F(S) is nothing but the closure of the one-dimensional stratum in the stable stratification of M = M(S). A complex point of F represents an algebraic curve which is a stable graph of copies of P 1 \ {0, 1, ∞}, save perhaps for an irreducible component of type (0, 4) or [START_REF] Anderson | Exactness properties of profinite completion functors[END_REF][START_REF] Anderson | Exactness properties of profinite completion functors[END_REF]. When d(S) = 1, that is d -1 = 0, F coincides with M. As soon as d(S) > 1, F is contained in the boundary ∂M and more generally F λ ⊂ ∂M λ . The importance of this one-dimensional stratum in the stratification of M was first recognized in connection with a conjecture formulated by W.Fulton, hence the notation (see e.g. [START_REF] Gibney | Towards the ample cone of Mg[END_REF]).

Each irreducible component of F is (isomorphic to) a moduli space of dimension 1 and can be triangulated as above into two triangles. Lifting that triangulation to the corresponding Teichmüller space T = T (S) produces again a copy of the Farey tesselation F . It is bicolored and complex conjugacy permutes the colors of the triangles. On the other hand, for any level λ ∈ Λ, one gets a cover F λ → F, which ramifies at most over points representing curves with the maximal (= d) number of nodes (singularities), that is graphs of trinions. The triangulation of F thus lifts uniquely to F λ . Moreover, and this is where the connection between F(S) and C P (S) comes in, it is easily seen that F λ (S) is naturally isomorphic to C λ P (S) = C P (S)/Γ λ , after identifying as usual the Farey graph with the corresponding tesselation. In slightly more detail, it is enough to show this for the trivial level, and then lift the result to every λ ∈ Λ. Moreover this is a local assertion, in the sense that we can fix d -1 curves, after which we are reduced to the one-dimensional situation of the last subsection. We record this as Proposition 7.5. For every λ ∈ Λ, F λ is a compact stable orbifold curve (i.e. a complex one-dimensional proper D-M stack with nodal singularities and finite group of automorphisms). It is equipped with a natural bicolored tesselation, whose dual graph is isomorphic to C λ P (S) = C P (S)/Γ λ Note that in the one-dimensional case the curve F coincides with the compactified moduli space, so does not lie at the boundary. This is one specificity of that case. The second is that the two-dimensional cells of the complex C P (S) are then given by triangles only. As a result C P (S) is represented by the Farey tesselation of the attending Teichmüller space, namely the Poincaré upper half-plane.

Suppose now that we are given an inverse system of invertible simplicial maps (simplicial automorphisms) φ λ : C λ P (S) → C λ P (S) between finite graphs (λ ∈ Λ). Here the quotient C λ P (S) = C P (S)/Γ λ can be considered naively or as an orbifold ; ditto for the morphisms. From Proposition 7.5) that C λ P (S) = C P (S)/Γ λ is the dual graph of a natural bicolored tesselation of F λ = F λ (S) and locally, everything happens as in the one-dimensional case. Appealing again e.g. to [START_REF] Voevodskii | Equilateral triangulations of Riemann surfaces and curves over algebraic number fields[END_REF], we find that this determines a complex structure on F λ for every λ ∈ Λ (F 0 = F) and in fact these are again projective algebraic curves defined over Q. Note that we may restrict to λ large enough (say dominating an abelian level of level > 2) so that F λ is indeed a bona fide curve, as opposed to an orbifold curve (one-dimensional DM stack); one can accommodate the stack structure just as well. In fact this complex structure concides with the one inherited from the fact that F λ ⊂ ∂M λ is nothing but the closure of the one-dimensional stratum of the stratified variety (more correctly DM stack) M λ . To see this it is enough to consider the case of the ground level F 0 = F ⊂ ∂M, the point being that Γ acts isometrically on the Teichmüller space T equipped with the Teichmüller metric.

Returning to the curve F λ we can proceed as in §7.4 and arrive at the following statement:

Proposition 7.6. Given a simplicial automorphism φ λ : C λ P (S) → C λ P (S), it determines a real analytic map (with the same name) φ λ : F λ → F λ , which on every analytically irreducible component of the projective curve F λ is either holomorphic or antiholomorphic.

We leave it to the reader to further enhance this statement. Basically all the objects are algebraic and defined over numberfields and so are the morphisms, a priori possibly after twisting by complex conjugacy. Passing to Teichmüller space we get the classical Bers bordification T of the Teichmüller space T , that is the Teichmüller space of stable nodal curves. The action of Γ(S) extends to T with quotient T /Γ(S) = M. It is natural to define F as the lift of F to ∂T . Is is endowed with a natural bicolored triangulation whose dual represents the pants graph C

(1) P (S) but although the 2-skeleton of C P (S) is thus uniquely determined, it is not clear whether it has a geometric (i.e. analytic) counterpart.

Theorem 2 . 4 .

 24 If d(S) > 1, all automorphisms of Γ(S) are inertia preserving: Aut * (Γ(S)) = Aut(Γ(S)).

Lemma 2 . 6 .

 26 If ρ, σ ∈ C(S) are compatible simplices: C * (S ρ ) ∩ C * (S σ ) = C * (S ρ∪σ ). If they are not compatible, this intersection is empty. Here all graphs C * (S τ ) (τ ∈ C(S)) are considered as subgraphs of C * (S). This lemma has a number of equally obvious consequences. For instance C * (S ρ ) ⊂ C * (S σ ) if and only if σ ⊂ ρ. Let us now return to the connections between C * and C P . The inclusion C P ⊂ C * can be made more precise (cf. §A.8), given that two simplicial embeddings of F in C P (S) are either disjoint, or else intersect in a single vertex. Lemma 2.7. C * (S) is obtained from C P (S) by replacing every maximal copy of the Farey graph F = C P (S 0,4 ) = C P (S 1,1 ) inside C P (S) by a copy of the complete graph G = C * (S 0,4 ) = C * (S 1,1 ) associated to the vertices of the given Farey graph.

  (β) = S σ , S (γ) = S τ , for certain simplices ρ, σ, τ ∈ C(S) with |ρ| = |σ| = |τ | = d(S) -k + 1. Moreover, because α ∩ β = ∅ (resp. β ∩ γ = ∅) ρ and σ (resp. σ and τ ) are compatible simplices. So we can consider ρ ∩ σ and σ ∩ τ , with |ρ ∩ σ| = |σ ∩ τ | = d(S) -k.

Theorem 3 . 8 (

 38 [START_REF] Boggi | On the procongruence completion of the Teichmüller modular groups[END_REF], Thm. 5.1). For any k ∈ Ẑ \ {0} the map dk : Ľ(S) → Γ(S) is injective.

Proposition 4 . 4 .

 44 Let α, β ∈ Č(S) be two simplices. Then: i) If U ⊂ G α is an open subgroup of G α , the normalizer NΓ(U ) ⊂ Γ(S) coincides with the stabilizer Γα of α for the action of Γ(S) on Č(S). In particular NΓ(U ) = NΓ(G α ). Moreover the centralizer ZΓ(U ) has finite index in NΓ(G α ), the latter being an extension of a finite permutation group by the former.

Lemma 5 . 3 .

 53 Let α ∈ L(S) be a simple closed curve on S connected hyperbolic. Then the dual link L - Č(S) (α) is chain connected if and only if the discrete dual link L - C(S) (α) is (chain) connected.

Lemma 5 . 5 .

 55 A simplicial automorphism φ : Č(S) → Č(S ) preserves both Sep and N Sep, that is we have Sep(S) = Sep(S ) and N Sep(S) = N Sep(S ).

4 .

 4 Let again G α ⊂ Γ(S) denote the commutative subgroup topologically generated by the (pro)twists along the (pro)curves attached to the vertices of α, and let U λ α = G α ∩ Γλ (S). Then U λ α is open in G α and by Proposition 4.4, for a simplex β ∈ Č(S), the intersection U λ α ∩ U λ β is open in U λ α if and only if α ⊂ β. So given φ ∈ Aut * ( Γλ (S)) it makes senses to define φ = γ λ (φ) ∈ Aut * ( Γλ (S)) via the formula:

Proposition 7. 3 .

 3 Aut( Fst ) (P GL 2 (Z)).

Definition 7 . 4 .

 74 now define a curve, or rather a one-dimensional orbifold (D-M stack) F λ : Let S be hyperbolic connected of modular dimension d = d(S). The one-dimensional orbifold F(S) ⊂ M(S) is such that its (closed) points represent Riemann surfaces (curves) with at least d -1 nodes (quadratic singularities). For an arbitrary level λ ∈ Λ we let F λ (S) ⊂ M λ (S) denote the preimage of F(S) via the canonical projection M λ (S) → M(S).

Appendix A. Some definitions and known results

We have gathered here a number of definitions, most of which but not all are classical, and a number of results in the discrete setting, most of which but not all are used in the text. The point is simply to provide the reader with the basic notation and material, together with some more or less standard references.

A.1. A finite type is a pair (g, n) of non negative integers. Given a type, we let S = S g,n denote theunique up to diffeomorphism -differentiable surface of genus g with n deleted points. We occasionally write g(S) for the genus of S. The points can also be be considered as "holes", provided isotopies do not fix the boundary circles. A surface is of type (g, n) if it is diffeomorphic to S g,n . The Euler characteristic of S g,n is χ(S) = 2 -2g -n; the surface is hyperbolic if 2g -2 + n > 0.

A.2. Attached to a surface S of type (g, n) are the Teichmüller space T (S) and moduli space M(S). We restrict henceforth to hyperbolic surfaces. The Teichmüller space T (S) is noncanonically identified with the standard Teichmüller space T g,n associated with the given type. It has dimension d(S) = d g,n = 3g -3 + n, which we call the modular dimension of S or of the given type -we will often drop the adjective "modular". In turn M(S) is -again noncanonically -identified with M g, [n] , the moduli space of curves of the given type, with unlabelled marked points. We use brackets ([n]) when the points are unlabelled, that is are considered setwise. Note that to be consistent we should write S g,[n] rather than S g,n but we nevertheless retain the latter piece of notation for simplicity; also, T g,[n] = T g,n because the definition of Teichmüller space involves a marking, so in particular the choice of generators of the fundamental group of the model surface.

A.3. We let M od(S) = π 0 (Dif f (S)) denote the (extended) mapping class group of S, i.e. the group of isotopy classes of diffeomorphisms of S. The index 2 subgroup of orientation preserving isotopy classes is denoted M od + (S). More generally an upper + will mean orientation preservating. We write Γ(S) = M od + (S) and call it the (Teichmüller) modular group. It can be seen as the orbifold fundamental group of M(S) and as the Galois group of the orbifold unramified cover T (S)/M(S). So we have the tautological exact sequence:

The group Γ(S) is (noncanonically) isomorphic to Γ g,[n] , defined as the fundamental group of the complex orbifold M g, [n] . The group Γ g,[n] is centerfree, except for 4 low-dimensional exceptions, i.e. types (0, 4), (1, 1), (1, 2) and (2, 0). In the first case the center is Klein's Vierergruppe ( Z/2 × Z/2); in the other three cases the center is isomorphic to Z/2, generated by the (hyper)elliptic involution. We refer to any elementary text on the subject for more detail.

A.4. Permutations of points play a certain role in the theory. The moduli space of curves of genus g with n ordered points is denoted M g,n . The cover M g,n /M g,[n] is finite, orbifold unramified (stack étale) and Galois with group S n , the permutation group on n symbols. Let us detail one low dimensional example (or exception) which is mentioned in the text. The group Γ 1, [START_REF] Artin | Etale Homotopy[END_REF] has center Z isomorphic to Z/2 as mentioned above. It is the direct product of that center and the corresponding ordered group: Γ 1, [START_REF] Artin | Etale Homotopy[END_REF] = Γ 1,2 × Z. Moreover Γ 1,2 ⊂ Γ 0, [START_REF] Boggi | Galois coverings of moduli spaces of curves and loci of curves with symmetry[END_REF] is the subgroup which corresponds to the permutations stabilizing the -say -fifth point. Geometrically speaking, to a genus 1 curve with 2 marked points one can associate 5 points, namely the 4 Weierstrass points plus the orbit of the two points under the elliptic involution; the two points can be indeed made to form an orbit, after a suitable translation. The 4 points can be permuted but the fifth one should be kept labeled under the action of the modular group, hence the above description. Finally, it is useful to note that Γ 1,2 is self-normalizing in Γ 0, [START_REF] Boggi | Galois coverings of moduli spaces of curves and loci of curves with symmetry[END_REF] , so in particular not normal.

A.5. We now briefly summarize the definitions pertaining to various curve complexes, referring to any of the many references (e.g. [START_REF] Ivanov | Subgroups of Teichmüller modular groups[END_REF][START_REF] Ivanov | Automorphisms of complexes of curves and of Teichmüller spaces[END_REF][START_REF] Luo | Automorphisms of the complex of curves[END_REF] etc.) for more detail. It is remarkable that we will actually need only consider graphs (and prographs), that is complexes of dimension 1.

Given a surface S, hyperbolic and of finite type (see §A1), we let L(S) denote the set of isotopy classes of simple closed curves on S not isotopic to boundary curves (circles around the marked points). A multicurve is a set of non intersecting elements of L where non intersecting means that there exist representatives which do not intersect (see [START_REF] Fathi | Travaux de Thurston sur les surfaces (seconde édition)[END_REF] or again any standard reference for detail).

The first complex C(S) is the one originally defined by W.J.Harvey in the late sixties. A k-simplex of C(S) is defined by a multicurve α = (α 0 , . . . , α k ), so that the vertices of C(S) correspond to elements of L(S). Boundary and face operators are defined by deletion and inclusion of curves respectively. This makes C(S) into a (non locally finite) simplicial complex of dimension d(S)-1 where d(S) is the modular dimension of S (see §A.2). We will write C (k) (S) for the k-dimensional skeleton of C(S) and use a similar notation for the other complexes. Note that L(S) = C (0) (S) is just the 0-skeleton (vertex set) of C(S) but it is nonetheless useful to retain a specific piece of notation.

There is a natural action of Γ(S) on C(S) determined by saying that to g ∈ Γ and a curve α ∈ L one associates g • α, the image of the curve by g, everything up to isotopy.

A.6. Next we define C G (S), the group theoretic complex. It is useful essentially in the complete case (see below), so is included in the present discrete setting essentially to fix notation. Here all objects pertain to the discrete topology, so we add a superscript "disc". Let Γ = Γ(S) and G disc (Γ) denote the set of all subgroups of Γ. To every simplex (i.e. multicurve) σ ∈ C(S) we assign the (discrete) free abelian group G disc σ ∈ G(Γ) spanned by the (Dehn) twists associated to σ. We then use the boundary and face operators as for C(S) in order to make G disc (Γ) into a simplicial complex, indeed a Boolean lattice.

In the discrete setting, C G (S) is (more or less trivially) isomorphic to C(S) and we define a Γ-action on C G (S) so as to make the natural isomorphism equivariant. To α ∈ L, that is a vertex of C(S), one thus assigns the cyclic group generated by τ α , the twist along α. Note that at this point, we should and do fix an orientation for S. Then for g ∈ Γ one has the well-known formula: τ g•α = gτ α g -1 ∈ Γ. The right-hand side of this equality defines an action of Γ on C G (S) which makes the natural isomorphism between C(S) and C G (S) Γ-equivariant. A.7. We then come to the pants complex C P (S). It was briefly mentioned the appendix of the classical 1980 paper by A.Hatcher and W.Thurston (see [START_REF] Hatcher | On the Teichmüller tower of mapping class groups[END_REF] or [START_REF] Margalit | Automorphisms of the pants complex[END_REF]) and first studied in [START_REF] Hatcher | On the Teichmüller tower of mapping class groups[END_REF] where it is shown to be connected and simply connected. It is a two dimensional, not locally finite complex whose vertices are given by the pants decomposition (i.e. maximal multicurves) of S; these correspond to the simplices of highest dimension (= d(S) -1) of C(S). Given two vertices α, α ∈ C P (S), they are connected by an edge if and only if α and α have d(S) -1 curves in common, so that up to relabelling (and of course isotopy) α i = α i , i = 1, . . . , d(S) -1, whereas α 0 and α 0 differ by an elementary move, which means the following. Cutting S along the α i 's, i > 0, there remains a surface Σ of modular dimension 1, so Σ is of type (1, 1) or (0, 4). Then α 0 and α 0 , which are supported on Σ, should intersect in a minimal way, that is they should have geometric intersection number 1 in the first case, and 2 in the second case (in the latter case their algebraic intersection number is 0). In the first case (genus 1), the edge (and move) is said to be of type S (for "simple", see [START_REF] Hatcher | On the Teichmüller tower of mapping class groups[END_REF]); in the second case (genus 0) of type A (for "associativity", see [START_REF] Hatcher | On the Teichmüller tower of mapping class groups[END_REF]). For d(S) = 1, the 1-skeleton of C P (S) is the Farey graph F .

We have thus defined the 1-skeleton C

P (S) of C P (S) which, following [START_REF] Margalit | Automorphisms of the pants complex[END_REF], we call the pants graph of S. We omit here the definition of the 2-cells of the complex C P (S) (see [START_REF] Hatcher | On the Teichmüller tower of mapping class groups[END_REF] or [START_REF] Margalit | Automorphisms of the pants complex[END_REF]), which enters in §7 only. They describe certain relations between elementary moves, that is they can be considered as elementary homotopies; as mentioned above, pasting them in makes C P (S) simply connected (cf. [START_REF] Hatcher | On the Teichmüller tower of mapping class groups[END_REF]). It is shown in [START_REF] Margalit | Automorphisms of the pants complex[END_REF] how to recover the full 2-dimensional pants complex from the pants graph. For d(S) = 1 the pants complex is the Farey tesselation, which we again denote F . We usually use only the pants graph (except in section 7), i.e. the 1-skeleton C

(1) P (S) of C P (S), which in order to simplify notation we often abusively denote C P (S).

A.8. We finally define the graph C * (S) which plays an important role in the complete case, while actually clarifying a number of issues even in the discrete case (see §2). The graph C * (S) shares the same set of vertices as C P (S), namely the maximal multicurves (a.k.a. pants decomposition) of S. The edges are defined simply by relaxing the minimal intersection condition in the definition of the edges of C P (S). In other words two vertices represented by maximal multicurves α = (α i ) i and α = (α i ) i (i = 0, . . . , d(S) -1) are joined by an edge if up to relabelling α i = α i for i > 0; then α 0 and α 0 lie on a surface of type (0, 4) or (1, 1). So C P (S) ⊂ C * (S) is a subgraph with the same set of vertices.

If S is connected (see however §A.9 below) of dimension 0, it is of type (0, 3) (a trinion or pair of pants); by convention, C P (S) = C * (S) is reduced to a point with no edge attached; note that usually one defines C(S 0,3 ) = ∅. If S is connected of dimension 1, it is of type (0, 4) or (1, 1). In both cases C P (S) = F coincides with the Farey graph. On the other hand, it is easily checked that C * (S) is the complete graph with the same vertices as F , which we denote by G: two simple closed curves on a surface of (modular) dimension 1 always intersect nontrivially. If d(S) > 1, C * (S) is nothing but the 1-skeleton of C(S) * , the complex dual to C(S). For this reason, when d(S) = 1, it becomes natural to define C(S) as the dual of G, which is not the usual convention but seems to be the right one for our purposes.

A.9. It is useful to extend the definitions of the graphs C P (S) and C * (S) to non connected surfaces. The extension is rather trivial yet it shows that these two graphs are particularly well-behaved. The definitions are simply unchanged. We will write V (S) for the set of vertices common to C * (S) and C P (S) (i.e. maximal multicurves), E(S) (resp. E P (S)) for the edges of C * (S) (resp. C P (S)): E P (S) ⊂ E(S).

Let S = S S be given as the disjoint sum of S and S , which themselves need not be connected. First note that modular dimension is additive: d(S) = d(S ) + d(S ). Then it is easy to describe C * (S) and C P (S) in terms of the graphs associated to S and S . For the vertices we get: V (S) = V (S ) × V (S ); and for the edges of C * (S): E(S) = E(S ) × V (S ) V (S ) × E(S ). Simply change E into E P for the case of C P . These prescriptions immediately generalize to an arbitrary number r of not necessarily connected pieces.

replace again E with E P when dealing with C P . A.10. We now come to completions, first of groups, then of the various simplicial complexes. Given S hyperbolic of finite type we start by indexing the inverse system of the cofinite (i.e. finite index) subgroups of Γ = Γ(S) Γ g,[n] by a set M , so that to any λ ∈ M there correspond a subgroup Γ λ and a cover M λ /M which we call a level structure following a traditional terminology in this context. For λ, µ ∈ M we write µ ≥ λ if Γ µ ⊆ Γ λ i.e. if M µ is a covering of M λ , and we say that M µ (resp. Γ µ ) dominates M λ (resp. Γ λ ).

For any subinverse system Λ ⊂ M we get the corresponding completion of Γ as the limit :

The (full) profinite completion is obtained when Λ = M and is denoted with a hat as usual:

The analogous definition can be given for any group. Note that the groups we consider, "arising from geometry", are discrete and finitely generated. It implies that the system of all invariant (a.k.a. characteristic) subgroups is cofinal. That is for any λ ∈ M one can find a (cofinite) invariant subgroup contained in Γ λ . The procongruence (or simply congruence) completion is specific of the situation at hand, selecting a particular subsystem Λ of cofinite subgroups of Γ. Denote by π = π 1 (S) the fundamental group of the surface S with respect to some basepoint and let K ⊂ π be a characteristic subgroup of π. The elements g ∈ Γ act on π (as 'mapping classes') up to inner automorphism, so there is a natural map: Γ → Out(π/K). We denote the kernel by Γ K ⊂ Γ and call it a principal congruence subgroup. It is normal and cofinite since π/K (and thus also Out(π/K)) is a finite group. A congruence subgroup of Γ is one which contains a principal subgroup. In particular, for m ≥ 2 a positive integer, the abelian level M (m) is defined by the subgroup Γ (m) which is the kernel of the natural map Γ → Sp 2g (Z/m), that is Γ (m) is the group of diffeomorphisms of S (considered modulo isotopy) which fix the homology of the associated unmarked or compact surface modulo m.

The congruence completion, denoted Γ, is obtained by choosing for Λ ⊂ M the system of all the congruence subgroups. We have a natural surjective map : Γ → Γ and the congruence conjecture (first proposed by N.Ivanov) asserts that this is actually an isomorphism, which amounts to stating that the congruence subgroups form a cofinal system in M . If true and vindicated, that is if indeed Γ = Γ, all the results of P.LOCHAK the present paper naturally come to hold true in the (full) profinite setting. For more on the congruence property, including from a homotopical viewpoint, and for references, we refer again to [START_REF] Lochak | Results and conjectures in profinite Teichmüller theory[END_REF].

A.11. We now come to profinite complexes of curves. More generally, let X • be a simplicial complex endowed with an action of Γ = Γ(S). Then we can define its profinite completion as the inverse limit:

which we regard as a simplicial object in the category of profinite sets. The above definition would of course be valid for other groups than Γ and spaces X which are not necessarily simplicial complexes. However the action of Γ on X has to satisfy certain geometric conditions which in our cases are easily met (see [START_REF] Boggi | Profinite Teichmüller theory[END_REF], §5). We apply the above to L(S), C(S), C P (S) and C * (S), obtaining the respective (full profinite) completions L(S), Ĉ(S), ĈP (S) and Ĉ * (S). We dropped the bullet subscript from the notation but stress that these are indeed simplicial objects. The profinite set L(S) is thus the set of procurves and it is the set of vertices of Ĉ(S). The complexes ĈP (S) and Ĉ * (S) are in fact prographs. We will often drop the prefix "pro" for simplicity but it should definitely be emphasized that these profinite spaces are complicated objects, just like profinite groups and even more so; note that the group completion Γ is obtained via the above procedure by letting Γ act on itself by translation. We refer to [START_REF] Boggi | Profinite Teichmüller theory[END_REF] for basic properties of these profinite complexes of curves.

In the present paper however we almost only use the congruence completion, obtained by replacing as above (see A.10) the full system M by the substem Λ of the congruence subroups. This procedure delivers the respective congruence completions, namely Ľ(S), Č(S), ČP (S) and Č * (S). The main reference is [START_REF] Boggi | On the procongruence completion of the Teichmüller modular groups[END_REF].

A.12. Our last item will deal very briefly with automorphisms of discrete modular groups and curve complexes. We refer to e.g. [START_REF] Ivanov | Automorphisms of complexes of curves and of Teichmüller spaces[END_REF][START_REF] Luo | Automorphisms of the complex of curves[END_REF] for more detailed statements and proofs. Our statements are geared towards the complete case and we have extracted what seems to be the significant minimum in that direction (more can be found in the body of the text). We let S be connected hyperbolic and of finite type; we assume that d(S) > 1 and S is not of type [START_REF] Anderson | Exactness properties of profinite completion functors[END_REF][START_REF] Artin | Etale Homotopy[END_REF], that is S is of type (0, 5) or d(S) > 2. This last assumption we make simply in order to avoid discussing well-known low-dimensional peculiarities (see [START_REF] Luo | Automorphisms of the complex of curves[END_REF]).

Then the automorphisms of the curve complex are described by the exact sequence:

where, in view of the profinite case, the group Z/2 should be considered as generated by complex conjugacy, so isomorphic to the Galois group Gal(C/R). With our assumptions Inn(Γ(S)) = Γ(S) except if S is of type (2, 0), in which case the center has order 2. Yet it is best to think of the left-hand group as Inn(Γ(S)) ⊂ Aut(Γ(S)).

Denoting by C (1) (S) the 1-skeleton of C(S), there is a natural injective map Aut(C(S)) → Aut(C (1) (S)) and this map is actually an isomorphism. This is an easy result, coming from a graph-theoretic characterization of the simplices of C(S) inside the graph C (1) (S): they are in one-to-one correspondence with the finite complete subgraphs, so have to be preserved by any automorphism of the graph.

Using the sequence (A 2) it is fairly easy to derive a description of the group automorphisms in the form of the following exact sequence:

In other words Aut(Γ(S)) = M od(S), Out(Γ(S)) Z/2 and the only non inner automorphism is generated by a reflection of the surface, that is an orientation reversing involution of the surface S, alias complex conjugacy, the generator of Gal(C/R). Note that the existence of such a reflection shows that the three sequences (A 1), (A 2) and (A 3) are split. Using (A 2) again, it is fairly easy to extend the above to any finite index subgroup of Γ(S) (cf. [START_REF] Ivanov | Automorphisms of complexes of curves and of Teichmüller spaces[END_REF]), quite a substantial improvement. In fact (A 3) remains valid if one replaces the middle group Γ(S) by a normal finite index subgroup Γ λ without changing the left and right hand groups. Put somewhat differently, there is an a priori injective map Aut(Γ(S)) → Aut(C(S)) and (A 3) asserts it is an isomorphism. This is a close analog of a famous result of Tits which states hat under suitable assumptions, the automorphisms of the building of an algebraic group come from the automorphisms of the group itself.