Daciberg Lima

Gonc ¸alves

John Guaschi
email: john.guaschi@unicaen.fr

Oscar Ocampo
email: oscaro@ufba.br

Carolina De
email: carolinapereiro@gmail.com

Miranda E Pereiro

CRYSTALLOGRAPHIC GROUPS AND FLAT MANIFOLDS FROM SURFACE BRAID GROUPS

Keywords: 2010 Mathematics Subject Classification. Primary: 20F36, 20H15; Secondary: 57N16 Surface braid groups, crystallographic group, flat manifold, Anosov diffeomorphism, Kähler manifold

Let M be a compact surface without boundary, and n ≥ 2. We analyse the quotient group B n (M )/Γ 2 (P n (M )) of the surface braid group B n (M ) by the commutator subgroup Γ 2 (P n (M )) of the pure braid group P n (M ). If M is different from the 2-sphere S 2 , we prove that B n (M )/Γ 2 (P n (M ))

If M is orientable, we prove a number of results regarding the structure of B n (M )/Γ 2 (P n (M )). We characterise the finite-order elements of this group, and we determine the conjugacy classes of these elements. We also show that there is a single conjugacy class of finite subgroups of B n (M )/Γ 2 (P n (M )) isomorphic either to S n or to certain Frobenius groups. We prove that crystallographic groups whose image by the projection B n (M )/Γ 2 (P n (M )) -→ S n is a Frobenius group are not Bieberbach groups. Finally, we construct a family of Bieberbach subgroups G n,g of B n (M )/Γ 2 (P n (M )) of dimension 2ng and whose holonomy group is the finite cyclic group of order n, and if X n,g is a flat manifold whose fundamental group is G n,g , we prove that it is an orientable Kähler manifold that admits Anosov diffeomorphisms.

Introduction

The braid groups of the 2-disc, or Artin braid groups, were introduced by Artin in 1925 and further studied in 1947 [START_REF] Artin | Theorie der Zöpfe[END_REF][START_REF] Artin | Theory of braids[END_REF]. Surface braid groups were initially studied by Zariski [START_REF] Zariski | The topological discriminant group of a Riemann surface of genus p[END_REF], and were later generalised by Fox and Neuwirth to braid groups of arbitrary topological spaces using configuration spaces as follows [START_REF] Fox | The braid groups[END_REF]. Let M be a compact, connected surface, and let n ∈ N. The nth ordered configuration space of M, denoted by F n (M), is defined by: F n (M) = {(x 1 , . . . , x n ) ∈ M n | x i = x j if i = j, i, j = 1, . . . , n} .

The n-string pure braid group P n (M) of M is defined by P n (M) = π 1 (F n (M)). The symmetric group S n on n letters acts freely on F n (M) by permuting coordinates, and the n-string braid group B n (M) of M is defined by B n (M) = π 1 (F n (M)/S n ). This gives rise to the following short exact sequence:

1 -→ P n (M) -→ B n (M) σ -→ S n -→ 1.
(1.1)

The map σ : B n (M) -→ S n is the standard homomorphism that associates a permutation to each element of S n . In [START_REF] Gonçalves | A quotient of the Artin braid groups related to crystallographic groups[END_REF][START_REF] Gonçalves | Almost-crystallographic groups as quotients of Artin braid groups[END_REF][START_REF] Gonçalves | Embeddings of finite groups in B n /Γ k (P n ) for k = 2[END_REF], three of the authors of this paper studied the quotient B n /Γ 2 (P n ), where B n is the n-string Artin braid group, P n is the subgroup of B n of pure braids, and Γ 2 (P n ) is the commutator subgroup of P n . In [START_REF] Gonçalves | A quotient of the Artin braid groups related to crystallographic groups[END_REF], it was proved that this quotient is a crystallographic group. Crystallographic groups play an important rôle in the study of the groups of isometries of Euclidean spaces (see Section 2 for precise definitions, as well as [START_REF] Charlap | Bieberbach groups and flat manifolds[END_REF][START_REF] Dekimpe | Almost-Bieberbach groups: affine and polynomial structures[END_REF][START_REF] Wolf | Spaces of constant curvature[END_REF] for more details). Using different techniques, Marin extended the results of [START_REF] Gonçalves | A quotient of the Artin braid groups related to crystallographic groups[END_REF] to generalised braid groups associated to arbitrary complex reflection groups [START_REF] Marin | Crystallographic groups and flat manifolds from complex reflection groups[END_REF]. Beck and Marin showed that other finite non-Abelian groups, not covered by [START_REF] Gonçalves | Almost-crystallographic groups as quotients of Artin braid groups[END_REF][START_REF] Marin | Crystallographic groups and flat manifolds from complex reflection groups[END_REF], embed in B n /Γ 2 (P n ) [START_REF] Beck | Torsion subgroups of quasi-abelianized braid groups[END_REF].

In this paper, we study the quotient B n (M)/Γ 2 (P n (M)) of B n (M), where Γ 2 (P n (M)) is the commutator subgroup of P n (M), one of our aims being to decide whether it is crystallographic or not.

The group extension (1.1) gives rise to the following short exact sequence:

1 -→ P n (M)/Γ 2 (P n (M)) -→ B n (M)/Γ 2 (P n (M)) σ -→ S n -→ 1.

(1.2)

Note that if M is an orientable, compact surface of genus g ≥ 1 without boundary and n = 1 then B 1 (M)/[P 1 (M), P 1 (M)] is the Abelianisation of π 1 (M), and is isomorphic to Z 2g , so it is clearly a crystallographic group.

In Section 2, we recall some definitions and facts about crystallographic groups, and if M is an orientable, compact, connected surface of genus g ≥ 1 without boundary, we prove that B n (M)/Γ 2 (P n (M)) is crystallographic.

Proposition 1. Let M be an orientable, compact, connected surface of genus g ≥ 1 without boundary, and let n ≥ 2. Then there exists a split extension of the form:

1 -→ Z 2ng -→ B n (M)/Γ 2 (P n (M)) σ -→ S n -→ 1, (1.3)
where the holonomy representation ϕ : S n -→ Aut(Z 2ng ) is faithful and where the action is defined by (2.5). In particular, the quotient B n (M)/Γ 2 (P n (M)) is a crystallographic group of dimension 2ng and whose holonomy group is S n .

As for B n /[P n , P n ], some natural questions arise for B n (M)/Γ 2 (P n (M)), such as the existence of torsion, the realisation of elements of finite order and that of finite subgroups, their conjugacy classes, as well as properties of some Bieberbach subgroups of B n (M)/Γ 2 (P n (M)). In Theorem 2, we characterise the finite-order elements of B n (M)/Γ 2 (P n (M)) and their conjugacy classes, from which we see that the conjugacy classes of finite-order elements of B n (M)/Γ 2 (P n (M)) are in one-to-one correspondence with the conjugacy classes of elements of the symmetric group S n . The results of Theorem 2 lead to the following question: if H 1 and H 2 are finite subgroups of B n (M)/Γ 2 (P n (M)) such that σ(H 1 ) and σ(H 2 ) are conjugate in S n , then are H 1 and H 2 conjugate? For each odd prime p, we shall consider the corresponding Frobenius group, which is the semi-direct product Z p ⋊ Z (p-1)/2 , the action being given by an automorphism of Z p of order (p -1)/2. In Proposition 12 we show that the conclusion of Theorem 2 holds for subgroups of B 5 (M)/Γ 2 (P 5 (M)) that are isomorphic to the Frobenius group Z 5 ⋊ Z 2 .

In Section 3, we study some Bieberbach subgroups of B n (M)/Γ 2 (P n (M)) whose construction is suggested by that of the Bieberbach subgroups of B n /Γ 2 (P n ) given in [START_REF] Ocampo | Bieberbach groups and flat manifolds with finite Abelian holonomy from Artin braid groups[END_REF].

Theorem 3. Let n ≥ 2, and let M be an orientable surface of genus g ≥ 1 without boundary. Let G n be the cyclic subgroup (n, n -1, . . . , 2, 1) of S n . Then there exists a subgroup

G n,g of σ -1 (G n )/Γ 2 (P n (M)) ⊂ B n (M)/Γ 2 (P n (M)) that is a Bieberbach group of dimension 2ng whose holonomy group is G n . Further, the centre Z( G n,g ) of G n,g is a free Abelian group of rank 2g.
The conclusion of the first part of the statement of Theorem 3 probably does not remain valid if we replace the finite cyclic group G n by other finite groups. In this direction, if p is an odd prime, in Proposition 13, we prove that there is no Bieberbach subgroup H of B p (M)/[P p (M), P p (M)] for which σ(H) is the Frobenius group Z p ⋊ Z (p-1)/2 .

It follows from the definition that crystallographic groups act properly discontinuously and cocompactly on Euclidean space, and that the action is free if the groups are Bieberbach. Thus there exists a flat manifold X n,g whose fundamental group is the subgroup G n,g of Theorem 3. Motivated by results about the holonomy representation of Bieberbach subgroups of the Artin braid group quotient B n /[P n , P n ] whose holonomy group is a 2-group obtained in [START_REF] Ocampo | On Bieberbach subgroups of B n /[P n , P n ] and flat manifolds with cyclic holonomy Z 2 d[END_REF], in Section 3, we make use of the holonomy representation of G n,g given in (3.4) to prove some dynamical and geometric properties of X n,g . To describe these results, we recall some definitions.

If f : M -→ M is a self-map of a Riemannian manifold, M is said to have a hyperbolic structure with respect to f if there exists a splitting of the tangent bundle T (M) of the form T (M) = E s ⊕ E u such that Df : E s -→ E s (resp. Df : E u -→ E u ) is contracting (resp. expanding). Further, the map f is called Anosov if it is a diffeomorphism and M has hyperbolic structure with respect to f . The classification of compact manifolds that admit Anosov diffeomorphisms is a problem first proposed by Smale [START_REF] Smale | Differentiable dynamical systems[END_REF]. Anosov diffeomorphisms play an important rôle in the theory of dynamical systems since their behaviour is generic in some sense. Porteous gave a criterion for the existence of Anosov diffeomorphisms of flat manifolds in terms of the holonomy representation [20, Theorems 6.1 and 7.1] that we shall use in the proof of Theorem 4.

We recall that a Kähler manifold is a 2n-real manifold endowed with a Riemannian metric, a complex structure, and a symplectic structure that is compatible at every point. For more about such manifolds, see [START_REF] Szczepański | Geometry of crystallographic groups[END_REF]Chapter 7]. A finitely-presented group is said to be a Kähler group if it is the fundamental group of a closed Kähler manifold. We may now state Theorem 4.

Theorem 4. Let n ≥ 2, and let X n,g be a 2ng-dimensional flat manifold whose fundamental group is the Bieberbach group G n,g of Theorem 3. Then X n,g is an orientable Kähler manifold with first Betti number 2g that admits Anosov diffeomorphisms.

The proof of Theorem 4 depends mainly on the holonomy representation of the Bieberbach group G n,g , and makes use of the eigenvalues of the matrix representation and the decomposition of the holonomy representation in irreducible representations using character theory.

Finally, in Section 4, we prove in Proposition 17 that the conclusion of Proposition 1 no longer holds if M is the sphere S 2 or a compact, non-orientable surface without boundary. More precisely, if n ≥ 1 then B n (M)/Γ 2 (P n (M)) is not a crystallographic group.

Crystallographic groups and quotients of surface braid groups

In this section, we start by recalling some definitions and facts about crystallographic groups. If M is a compact, orientable surface without boundary of genus g ≥ 1, in Proposition 1, we prove that the quotient B n (M)/Γ 2 (P n (M)) is a crystallographic group that is isomorphic to Z 2ng ⋊ ϕ S n . We also determine the conjugacy classes of the finite-order elements of B n (M)/Γ 2 (P n (M)) in Theorem 2.

2.1. Crystallographic groups. In this section, we recall briefly the definitions of crystallographic and Bieberbach groups, and the characterisation of crystallographic groups in terms of a representation that arises in certain group extensions whose kernel is a free Abelian group of finite rank and whose quotient is finite. We also recall some results concerning Bieberbach groups and the fundamental groups of flat Riemannian manifolds. For more details, see [4, Section I.1.1], [5, Section 2.1] or [START_REF] Wolf | Spaces of constant curvature[END_REF]Chapter 3].

Let G be a Hausdorff topological group. A subgroup H of G is said to be discrete if it is a discrete subset. If H is a closed subgroup of G then the quotient space G/H admits the quotient topology for the canonical projection π : G -→ G/H, and we say that H is uniform if G/H is compact. From now on, we identify Aut(Z m ) with GL(m, Z). A discrete, uniform subgroup Π of R m ⋊O(m, R) ⊆ Aff(R m ) is said to be a crystallographic group of dimension m. If in addition Π is torsion free then Π is called a Bieberbach group of dimension m.

If Φ is a group, an integral representation of rank m of Φ is defined to be a homomorphism Θ : Φ -→ Aut(Z m ). Two such representations are said to be equivalent if their images are conjugate in Aut(Z m ). We say that Θ is a faithful representation if it is injective. We recall the following characterisation of crystallographic groups.

Lemma 5 ([10, Lemma 8]). Let Π be a group. Then Π is a crystallographic group if and only if there exists an integer m ∈ N, a finite group Φ and a short exact sequence of the form:

0 -→ Z m -→ Π ζ -→ Φ -→ 1, (2.1)
such that the integral representation Θ : Φ -→ Aut(Z m ) induced by conjugation on Z m and defined by Θ(ϕ)(x) = πxπ -1 for all x ∈ Z m and ϕ ∈ Φ, where π ∈ Π is such that ζ(π) = ϕ, is faithful.

If Π is a crystallographic group, the integer m, the finite group Φ and the integral representation Θ : Φ -→ Aut(Z m ) appearing in the statement of Lemma 5 are called the dimension, the holonomy group and the holonomy representation of Π respectively.

We now recall the connection between Bieberbach groups and manifolds. A Riemannian manifold M is called flat if it has zero curvature at every point. By the first Bieberbach Theorem, there is a correspondence between Bieberbach groups and fundamental groups of flat Riemannian manifolds without boundary (see [START_REF] Dekimpe | Almost-Bieberbach groups: affine and polynomial structures[END_REF]Theorem 2.1.1] and the paragraph that follows it). By [START_REF] Wolf | Spaces of constant curvature[END_REF]Corollary 3.4.6], the holonomy group of a flat manifold M is isomorphic to the group Φ. In 1957, Auslander and Kuranishi proved that every finite group is the holonomy group of some flat manifold (see [START_REF] Wolf | Spaces of constant curvature[END_REF]Theorem 3.4.8] and [4, Theorem III.5.2]). It is well known that a flat manifold determined by a Bieberbach group Π is orientable if and only if the integral representation Θ : Φ -→ GL(m, Z) satisfies Im(Θ) ⊆ SL(m, Z) [START_REF] Dekimpe | Almost-Bieberbach groups: affine and polynomial structures[END_REF]Theorem 6.4.6 and Remark 6.4.7]. This being the case, Π is said to be an orientable Bieberbach group.

2.2.

The group B n (M)/Γ 2 (P n (M)). Let M be a compact, orientable surface without boundary of genus g ≥ 1. Besides showing that the group B n (M)/Γ 2 (P n (M)) is crystallographic, we shall also be interested in the conjugacy classes of its elements by elements of P n (M)/Γ 2 (P n (M)), as well as the conjugacy classes of its finite subgroups. In order to study these questions, it is useful to have an algebraic description of this quotient at our disposal. We will make use of the presentations of the (pure) braid groups of M given in [13, Theorems 2.1 and 4.2], where for all 1 ≤ i < j ≤ n, 1 ≤ r ≤ 2g and 1 ≤ k ≤ n, the elements a k,r and T i,j in B n (M) are described in [START_REF] González-Meneses | New presentations of surface braid groups[END_REF]Figure 9], and for all 1 ≤ i ≤ n -1, the elements σ i are the classical generators of the Artin braid group that satisfy the Artin relations:

σ i σ j = σ j σ i for all 1 ≤ i, j ≤ n -1, |i -j| ≥ 2 σ i σ i+1 σ i = σ i+1 σ i σ i+1 for all 1 ≤ i ≤ n -2.
(2.2)

We recall that the full twist braid of B n (M), denoted by ∆ 2 n , is defined by: ∆

2 n = (σ 1 • • • σ n-1 ) n , (2.3)
and is equal to:

∆ 2 n = A 1,2 (A 1,3 A 2,3 ) • • • (A 1,n A 2,n • • • A n-1,n ), (2.4)
where for 1 ≤ i < j ≤ n, the elements A i,j are the usual Artin generators of P n defined by

A i,j = σ j-1 • • • σ i+1 σ 2 i σ -1 i+1 • • • σ -1 j-1 . By abuse of notation, in what follows, if α ∈ B n (M)
, we also denote its B n (M)/Γ 2 (P n (M))-coset by α. The following proposition gives some relations in B n (M) that will be relevant to our study of B n (M)/Γ 2 (P n (M)). Proposition 6. Let M be a compact, orientable surface without boundary of genus g ≥ 1, let . By [13, page 439], T j-1,j-1 = 1 for all 1 < j ≤ n + 1, and it follows from part (c) that T j-1,j ∈ Γ 2 (P n (M)) for all 2 ≤ j ≤ n, and then by induction on j -i that T i,j ∈ Γ 2 (P n (M)) for all 1 ≤ i ≤ j ≤ n. Using the Artin relations (2.2) and part (b), we have A i,j = T -1 i,j-1 T i,j for all 1 ≤ i < j ≤ n, so A i,j also belongs to Γ 2 (P n (M)) by part (c), and thus ∆ 2 n ∈ Γ 2 (P n (M)) by (2.4). This allows us to compute the Abelianisation of P n (M).

1 ≤ i ≤ n -1, 1 ≤ j ≤ n and 1 ≤ r ≤ 2g, and let A j,r = a j,1 • • • a j,r-1 a -1 j,r+1 • • • a -1 j,2g . The following relations hold in B n (M): (a) σ i a j,r σ -1 i =              a i+1,r σ -2 i if j = i and r is even σ 2 i a i+1,r if j = i and r is odd σ 2 i a i if j = i + 1 and r is even a i,r σ -2 i if j = i + 1 and r is odd a j,r if j = i, i + 1. (b) T i,j = σ i σ i+1 • • • σ j-2 σ 2 j-1 σ j-2 • • • σ i where 1 ≤ i, j ≤ n and i + 1 < j, and T i,i+1 = σ 2 i for all 1 ≤ i ≤ n -1. (c) T i,j = [a i,1 • • • a i,r , A j,r ] T i,j-1 , for all 1 ≤ i < j ≤ n and 1 ≤ r ≤ 2g. (d) For all 1 ≤ i ≤ j ≤ n, T i,j and ∆ 2 n belong to Γ 2 (P n (M)). Proof. Part (a)
Corollary 7. Let M be a compact, orientable surface without boundary of genus g ≥ 1, and let n ≥ 1. Then the Abelianisation of P n (M) is a free Abelian group of rank 2ng, for which {a i,r | i = 1, . . . , n and r = 1, . . . , 2g} is a basis.

Proof. The result follows from the presentation of P n (M) given in [13, Theorem 4.2] and the fact that for all 1 ≤ i < j ≤ n, T i,j ∈ Γ 2 (P n (M)) by Proposition 6(d).

For all 1 ≤ i ≤ n -1, we have σ(σ i ) = τ i , where τ i denotes the transposition (i, i + 1) in S n . Using Proposition 6(a), and identifying Z 2ng with P n (M)/Γ 2 (P n (M)) via Corollary 7, we obtain the induced action ϕ :

S n -→ Aut(Z 2ng ), that for all 1 ≤ i ≤ n -1, 1 ≤ j ≤ n and 1 ≤ r ≤ 2g, is defined by: ϕ(τ i )(a j,r ) = σ i a j,r σ -1 i = a τ i (j),r .
(2.5) The following result is the analogue of [10, Proposition 12] for braid groups of orientable surfaces. Proposition 8. Let M be a compact, orientable surface without boundary of genus g ≥ 1, and let n ≥ 1. Let α ∈ B n (M)/Γ 2 (P n (M)), and let π = σ(α -1 ). Then αa i,r α -1 = a π(i),r in B n (M)/Γ 2 (P n (M)) for all 1 ≤ i ≤ n and 1 ≤ r ≤ 2g.

Proof. The proof is similar to that of [START_REF] Gonçalves | A quotient of the Artin braid groups related to crystallographic groups[END_REF]Proposition 12], and makes use of (2.5). The details are left to the reader.

We now give a presentation of B n (M)/Γ 2 (P n (M)). Proposition 9. Let M be a compact, orientable surface without boundary of genus g ≥ 1, and let n ≥ 1. The quotient group B n (M)/Γ 2 (P n (M)) has the following presentation:

Generators: σ 1 , . . . , σ n-1 , a i,r , 1 ≤ i ≤ n, 1 ≤ r ≤ 2g. Relations: (a) the Artin relations (2.2). (b) σ 2 i = 1, for all i = 1, . . . , n -1. (c) [a i,r , a j,s ] = 1, for all i, j = 1, . . . , n and r, s = 1, . . . , 2g. (d) σ i a j,r σ -1 i = a τ i (j),r for all 1 ≤ i ≤ n -1, 1 ≤ j ≤ n and 1 ≤ r ≤ 2g.
Proof. The given presentation of B n (M)/Γ 2 (P n (M)) may be obtained by applying the standard method for obtaining a presentation of a group extension [14, Proposition 1, p. 139] to the short exact sequence (1.2) for M satisfying the hypothesis, and using Corollary 7, the equality σ(σ i ) = τ i for all i = 1, . . . , n -1, and the fact that the relations (a) and (b) constitute a presentation of S n for the generating set {τ 1 , . . . , τ n-1 }.

We may now prove Proposition 1.

Proof of Proposition 1. Assume that n ≥ 2. The short exact sequence (1.3) is obtained from (1.2) using Corollary 7. To prove that the short exact sequence (1.3) splits, let ψ : S n -→ B n (M)/Γ 2 (P n (M)) be the map defined on the generating set {τ 1 , . . . , τ n-1 } of S n by ψ(τ i ) = σ i for all i = 1, . . . , n -1.

Relations (a) and (b) of Proposition 9 imply that ψ is a homomorphism. Consider the action ϕ : S n -→ Aut(Z 2ng ) defined by (2.5). By Proposition 8, ϕ(θ) is the identity automorphism if and only if θ is the trivial permutation, from which it follows that ϕ is injective. The rest of the statement of Proposition 1 is a consequence of Lemma 5.

Corollary 10. Let M be a compact, orientable surface without boundary of genus g ≥ 1, let n ≥ 2, and let H be a subgroup of S n . Then the group σ -1 (H)/Γ 2 (P n (M)) is a crystallographic group of dimension 2ng whose holonomy group is H.

Proof. The result is an immediate consequence of Proposition 1 and [10, Corollary 10].

We now turn to the proof of Theorem 2. For this, we will require the following lemma.

Lemma 11. Let M be a compact, orientable surface without boundary of genus g ≥ 1, and let n ≥ 1.

Let z ∈ B n (M)/Γ 2 (P n (M)). Let z = ω n i=1 2g r=1 a s i,r i,r ∈ B n (M)/Γ 2 (P n (M))
, where ω = ψ(σ(z)), and s i,r ∈ Z for all 1 ≤ i ≤ n and 1 ≤ r ≤ 2g. Suppose that σ(z) is the m-cycle (l 1 , . . . , l m ), where m ≥ 2, and let k ∈ N be such that m divides k. Then

z k = n i=1 2g r=1 a t i,r
i,r where for all 1 ≤ i ≤ n and 1 ≤ r ≤ 2g, t i,r ∈ Z is given by:

t i,r = ks i,r if i / ∈ {l 1 , . . . , l m } k m Σ m j=1 s l j ,r if i = l j , where 1 ≤ j ≤ m.
(2.6) arises from (1.3). By Corollary 7 and Proposition 8 and using the fact that ω k = 1, we obtain:

Proof. Let z ∈ B n (M)/Γ 2 (P n (M))
z k = ω n i=1 2g r=1 a s i,r i,r k = ω k k j=1 ω j-k n i=1 2g r=1 a s i,r i,r ω k-j = k j=1 n i=1 2g r=1 a s i,r σ(ω k-j )(i),r = n i=1 2g r=1 k j=1 a s σ(ω j-k )(i),r i,r = n i=1 2g r=1 a t i,r i,r , (2.7)
where t i,r = k j=1 s σ(ω j-k )(i),r for all 1 ≤ i ≤ n and 1 ≤ r ≤ 2g. Equation (2.6) then follows, using the fact that σ(ω -u )(l i ) = l i-u for all u ∈ Z, where the index i -u is taken modulo m.

We now prove Theorem 2.

Proof of Theorem 2. (a) Let ψ : S n -→ B n (M)/Γ 2 (P n (M)) be the section for the short exact sequence (1.3) given in the proof of Proposition 1, and let e 1 and e 2 be finite-order elements of B n (M)/Γ 2 (P n (M)). If e 1 and e 2 are conjugate in B n (M)/Γ 2 (P n (M)) then σ(e 1 ) and σ(e 2 ) are conjugate in S n , and so have the same cycle type. Conversely, suppose that the permutations σ(e 1 ) and σ(e 2 ) have the same cycle type. Then they are conjugate in S n , so there exists ξ ∈ S n such that σ(e 1 ) = ξσ(e 2 )ξ -1 , and up to substituting e 2 by ψ(ξ -1 )e 2 ψ(ξ) if necessary, we may assume that σ(e 1 ) = σ(e 2 ). We claim that if θ is any finite-order element of B n (M)/Γ 2 (P n (M)) then θ and ψ(σ(θ)) are conjugate in B n (M)/Γ 2 (P n (M)). This being the case, for i = 1, 2, e i is conjugate to ψ(σ(e i )), but since ψ(σ(e 1 )) = ψ(σ(e 2 )), it follows that e 1 and e 2 are conjugate as required, which proves the first part of the statement. To prove the claim, set θ = ω n i=1 2g r=1 a s i,r i,r ∈ B n (M)/Γ 2 (P n (M)) as in the proof of Lemma 11, where ω = ψ(σ(θ)). It thus suffices to prove that θ and ω are conjugate in B n (M)/Γ 2 (P n (M)). Let σ(θ) = τ 1 • • • τ d be the cycle decomposition of σ(θ), where for i = 1, . . . , d, τ i = (l i,1 , . . . , l i,m i ) is an m i -cycle, and m i ≥ 2, and let k = lcm(m 1 , . . . , m d ) be the order of σ(θ), which is also the order of θ and of ω. For t = 1, . . . , n and r = 1, . . . , 2g, we define p t,r ∈ Z as follows. If t does not belong to the support Supp(σ(θ)) of σ(θ), let p t,r = 0. If t ∈ Supp(σ(θ)) then t = l i,q for some 1 ≤ i ≤ d and 1 ≤ q ≤ m i , and we define p t,r = -q j=1 s l i,j ,r .

It follows from Lemma 11 and the fact that θ is of order k that p l i,m i ,r = 0, for all i = 1, . . . , d. Then for all 1 ≤ i ≤ d, 2 ≤ q ≤ m i and 1 ≤ r ≤ 2g, we have: p l i,q-1 ,r -p l i,q ,r = s l i,q ,r and p l i,m i ,r -p l i,1 ,r = -p l i,1 ,r = s l i,1 ,r .

(2.8)

Let α = n i=1 2g r=1 a p i,r
i,r ∈ P n (M)/Γ 2 (P n (M)). By Corollary 7 and Proposition 8, we have:

αωα -1 = ω. ω -1 n i=1 2g r=1 a p i,r i,r ω. n i=1 2g r=1 a -p i,r i,r = ω n i=1 2g r=1 a p σ(ω -1 )(i),r i,r . n i=1 2g r=1 a -p i,r i,r = ω n i=1 2g r=1 a p σ(ω -1 )(i),r -p i,r i,r = ω n i=1 2g r=1 a s i,r i,r = θ,
where we have also made use of ( Since Ker (π) = Z n is torsion free, the restriction of π to s(H) is injective, and so π • s : H -→ S n is an isomorphism. Let us prove that the subgroups s(H) and S n of Z n ⋊ S n are conjugate. Let {τ 1 , . . . , τ n-1 } be the generating set of S n defined previously, and for i = 1, . . . , n -1, let α i be the unique element of H for which π • s(α i ) = τ i . Then H is generated by {α 1 , . . . , α n-1 }, subject to the Artin relations and α 2 i = 1 for all i = 1, . . . , n -1. There exist a i,j ∈ Z, where 1 ≤ i ≤ n -1 and 1 ≤ j ≤ n, such that s(α i ) = (a i,1 , . . . , a i,n )τ i in Z n ⋊ S n . Using the action of S n on Z n and the fact that α 2 i = 1, it follows that a i,j = 0 for all j = i, i + 1, and a i,i+1 = -a i,i . Then s(α i ) = (0, . . . , 0, a i , -a i , 0, . . . , 0)τ i for all 1 ≤ i ≤ n -1, where a i = a i,i , and the element a i is in the ith position. One may check easily that these elements also satisfy the Artin relations in Z n ⋊ S n . Let x 1 ∈ Z, and for i = 2, . . . , n, let x i = x 1 -i-1 j=1 a j . Thus x i -x i+1 = a i for all i = 1, . . . , n -1, and so (x 1 , x 2 , . . . , x i , x i+1 , . . . , x n )τ i (-x 1 , -x 2 , . . . , -x i , -x i+1 , . . . -x n ) = (0, . . . , 0, a i , -a i , 0, . . . , 0)τ i = s(α i ). We conclude that the subgroups s(H) and S n of Z n ⋊ S n are conjugate.

Returning to the general case where H embeds in (Z n ⊕ • • • ⊕ Z n ) ⋊ S n , the previous paragraph shows that the embedding of H into each Z n ⋊ S n is conjugate by an element of the same factor Z n to the factor S n . The result follows by conjugating by the element of Z n ⊕ • • • ⊕ Z n whose ith factor is the conjugating element associated to the ith copy of Z n ⋊ S n for all i = 1, . . . , 2g.

With the statement of Theorem 2 in mind, one may ask whether the result of the claim extends to other finite subgroups. More precisely, if G is a finite subgroup of B n (M)/Γ 2 (P n (M)), are G and ψ(σ(G)) conjugate? We have a positive answer in the following case.

Proposition 12. Let M be a compact, orientable surface without boundary of genus g ≥ 1. If H 1 and H 2 are subgroups of B 5 (M)/Γ 2 (P 5 (M)) that are isomorphic to the Frobenius group Z 5 ⋊ Z 2 then they are conjugate.

Proof. Using Proposition 1, we identify B 5 (M)/Γ 2 (P 5 (M)) with Z 10g ⋊ S 5 . As in the proof of Theorem 2(b), we decompose the first factor as a direct sum Z 10g = Z 5 ⊕ • • • ⊕ Z 5 of 2g copies of Z 5 , which we interpret as a Z[S 5 ]-module, the module structure being given by Proposition 8.

Let H be a group isomorphic to a subgroup of S 5 that embeds in (Z 5 ⊕ • • • ⊕ Z 5 ) ⋊ S 5 ∼ = B 5 (M)/Γ 2 (P 5 (M)). Using the Z[S 5 ]-module structure of Z 10g given above, it follows that H embeds in Z 5 ⋊ S 5 , for any one of the 2g summands of Z 5 . We will first prove the statement for the embedding of the Frobenius group Z 5 ⋊ Z 2 in Z 5 ⋊ S 5 , and then deduce the result in the general case. Let H be this Frobenius group, let s : H -→ Z 5 ⋊ S 5 be an embedding, let π : Z 5 ⋊ S 5 -→ S 5 be projection onto the first factor, and let ψ : S 5 -→ Z 5 ⋊ S 5 be inclusion into the second factor. Since Ker (π) = Z 5 is torsion free, the restriction of π to s(H) is injective, and so π • s : H -→ S 5 is an embedding of H into S 5 . Let us prove that the subgroups s(H) and ψ • π • s(H) of Z 5 ⋊ S 5 are conjugate. First observe that the Frobenius group Z 5 ⋊ Z 2 embeds in S 5 by sending a generator ι 5 of Z 5 to the permutation w 1 = (1, 2, 3, 4, 5) and the generator ι 2 of Z 2 to w 2 = (1, 4)(2, 3). The group S 5 contains six cyclic subgroups of order 5 that are conjugate to w 1 , from which we deduce the existence of six pairwise conjugate subgroups of S 5 isomorphic to Z 5 ⋊ Z 2 , each of which contains one of the cyclic subgroups of order 5. We claim that these are exactly the subgroups of S 5 isomorphic to Z 5 ⋊ Z 2 . To see this, let K be such a subgroup. Then the action of an element k of K of order 2 on a 5-cycle (a 1 , . . . , a 5 ) of K inverts the order of the elements a 1 , . . . , a 5 , and this can only happen if k is a product of two disjoint transpositions. Further, there are exactly five products of two disjoint transpositions whose action by conjugation on (a 1 , . . . , a 5 ) inverts the order of the elements a 1 , . . . , a 5 , and these are precisely the elements of K of order 2. In particular, each cyclic subgroup of S 5 of order 5 is contained in exactly one subgroup of S 5 isomorphic to Z 5 ⋊ Z 2 . This proves the claim. So up to composing π by an inner automorphism of S 5 if necessary, we may assume that π • s(H) = w 1 , w 2 . Applying methods similar to those of the proof of Lemma 11, the relations ι 5 5 = 1 and ι 2 2 = 1 imply that there exist a 1 , . . . , a 5 , x, y ∈ Z such that:

s(w 1 ) = (a 1 , a 2 , a 3 , a 4 , -a 1 -a 2 -a 3 -a 4
)w 1 and s(w 2 ) = (x, y, -y, -x, 0)w 2 .

(2.9)

Taking the image of the relation

w 2 w 1 w -1 2 = w -1
1 by s, using (2.9) and simplifying the resulting expression, we obtain:

x = -a 2 -a 3 -a 4 and y = -a 3 .

(2.10)

Any map s : H -→ Z 5 ⋊ S 5 of the form (2.9) for which the relations (2.10) hold gives rise to an embedding of H in Z 5 ⋊ S 5 . We claim that the image of the embedding is conjugate to the group w 1 , w 2 (viewed as a subgroup of the second factor of Z 5 ⋊ S 5 ). To do so, let s : H -→ Z 5 ⋊ S 5 of the form (2.9) for which the relations (2.10) hold. We will show that there exists a ∈ Z 5 such that aw i a -1 = s(w i ) for i = 1, 2. Let λ 5 ∈ Z, and for i = 1, . . . , 4, let λ i = λ 5 + i j=1 a i , and let a = (λ 1 , λ 2 , λ 3 , λ 4 , λ 5 ) ∈ Z 5 . Then in Z 5 ⋊ S 5 , using (2.9) and (2.10), we have:

aw 1 a -1 = (λ 1 , λ 2 , λ 3 , λ 4 , λ 5 )w 1 (-λ 1 , -λ 2 , -λ 3 , -λ 4 , -λ 5 ) = (λ 1 , λ 2 , λ 3 , λ 4 , λ 5 ). (-λ 5 , -λ 1 , -λ 2 , -λ 3 , -λ 4 )w 1 = (λ 1 -λ 5 , λ 2 -λ 1 , λ 3 -λ 2 , λ 4 -λ 3 , λ 5 -λ 4 )w 1 = (a 1 , a 2 , a 3 , a 4 , -a 1 -a 2 -a 3 -a 4 )w 1 = s(w 1 ),
and

aw 2 a -1 = (λ 1 , λ 2 , λ 3 , λ 4 , λ 5 )w 2 (-λ 1 , -λ 2 , -λ 3 , -λ 4 , -λ 5 ) = (λ 1 , λ 2 , λ 3 , λ 4 , λ 5 ). (-λ 4 , -λ 3 , -λ 2 , -λ 1 , -λ 5 )w 2 = (λ 1 -λ 4 , λ 2 -λ 3 , λ 3 -λ 2 , λ 4 -λ 1 , 0)w 2 = (-a 2 -a 3 -a 4 , -a 3 , a 3 , a 2 + a 3 + a 4 , 0)w 2 = (x, y, -y, -x, 0)w 2 = s(w 2 ).
It follows that the subgroups s(H) and w 1 , w 2 are conjugate in Z 5 ⋊ S 5 .

As in the proof of Theorem 2(b), returning to the general case where H embeds in (Z 5 ⊕ • • • ⊕ Z 5 ) ⋊ S 5 , the previous paragraph shows that the embedding of H into each Z 5 ⋊ S 5 is conjugate by an element of the same factor Z 5 to the factor S 5 . The result follows by conjugating by the element of Z 5 ⊕ • • • ⊕ Z 5 whose ith factor is the conjugating element associated to the ith copy of Z 5 ⋊ S 5 for all i = 1, . . . , 2g.

3. Some Bieberbach subgroups of B n (M)/Γ 2 (P n (M)) and Kähler flat manifolds By Corollary 10, for any subgroup H of S n , the quotient group σ -1 (H)/Γ 2 (P n (M)) is a crystallographic group that is not Bieberbach since it has torsion elements. We start by proving Theorem 3, which states that B n (M)/Γ 2 (P n (M)) admits Bieberbach subgroups. More precisely, for all n ≥ 2, we will consider the cyclic subgroup G n = (n, n -1, . . . , 2, 1) of S n , and we show that σ -1 (G n )/Γ 2 (P n (M)) admits a Bieberbach subgroup G n,g of dimension 2ng whose holonomy group is G n . The group G n,g is thus the fundamental group of a flat manifold. In Theorem 4, we will prove that this flat manifold is orientable and admits a Kähler structure as well as Anosov diffeomorphisms. Proposition 6(d). By Proposition 8, the action by conjugation of α n-1 on the elements of the basis {a i,r | i = 1, . . . , n and r = 1, . . . , 2g} of P n (M)/Γ 2 (P n (M)) given by Corollary 7 is as follows:

Proof of Theorem 3. Let α n-1 = σ 1 • • • σ n-1 ∈ B n (M)/Γ 2 (P n (M)). By equation (1.3), σ(α n-1 ) gen- erates the subgroup G n of S n . Further, the full twist of B n (M) is a coset representative of α n n-1 by (2.3), hence α n n-1 = 1 in B n (M)/Γ 2 (P n (M)) using
α n-1 : a 1,r -→ a 2,r -→ • • • -→ a n-1,r -→ a n,r -→ a 1,
r for all r = 1, . . . , 2g.

(

Using (3.1) and the fact that α n n-1 = 1 in B n (M)/Γ 2 (P n (M)), we have:

(a 1,1 α n-1 ) n = a 1,1 a 2,1 • • • a n,1 (α n-1 ) n = a 1,1 a 2,1 • • • a n,1 ∆ 2 n = n i=1 a i,1 . (3.2)
Let X = {a 1,1 α n-1 , a n i,r | 1 ≤ i ≤ n and 1 ≤ r ≤ 2g}, let Y = { n i=1 a i,1 , a n i,r | 1 ≤ i ≤ n and 1 ≤ r ≤ 2g}
, and let G n,g (resp. L) be the subgroup of σ -1 (G n )/Γ 2 (P n (M)) generated by X (resp. Y ). Then the restriction σ Gn,g : G n,g -→ G n is surjective, and using (3.2), we see that L is a subgroup of G n,g .

We claim that L = Ker σ Gn,g . Clearly, L ⊂ Ker σ Gn,g . Conversely, let w ∈ Ker σ Gn,g . Writing w in terms of the generating set X of G n,g and using (3.1) and Corollary 7, it follows that w = (a 1,1 α n-1 ) m 1≤i≤n 1≤r≤2g a nδ i,r i,r , where m ∈ Z and δ i,r ∈ Z for all 1 ≤ i ≤ n and 1 ≤ r ≤ 2g. The fact that w ∈ Ker σ Gn,g implies that n divides m, and so w = ((a 1,

1 α n-1 ) n ) m/n 1≤i≤n 1≤r≤2g a nδ i,r i,r
∈ L, which proves the claim. Thus the following extension:

1 -→ L -→ G n,g σ Gn,g -→ G n -→ 1, (3.3)
is short exact. Now L is also a subgroup of P n (M)/Γ 2 (P n (M)), which is free Abelian of rank 2ng by Corollary 7. Since {a i,r | i = 1, . . . , n and r = 1, . . . , 2g} is a basis of

P n (M)/Γ 2 (P n (M)), it follows from analysing Y that Y ′ = { n i=1 a i,1 , a n i,1 , a n j,r | 2 ≤ i ≤ n, 1 ≤ j ≤ n and 2 ≤ r ≤ 2g} is a basis of L.
In particular L is free Abelian of rank 2ng. In terms of the basis Y ′ , the holonomy representation ρ : G n -→ Aut(L) associated with the short exact sequence (3.3) is given by the block diagonal matrix:

ρ((1, n, n -1, . . . , 2)) =   M 1 M 2 . . . M 2g   , (3.4)
where M 1 , . . . , M 2g are the n-by-n matrices satisfying:

M 1 =     1 0 0 ••• 0 0 n 0 0 0 ••• 0 0 -1 0 1 0 ••• 0 0 -1 0 0 1 ••• 0 0 -1 . . . . . . . . . . . . . . . . . . . . . 0 0 0 ••• 1 0 -1 0 0 0 ••• 0 1 -1     and M 2 = • • • = M 2g =     0 0 0 ••• 0 0 1 1 0 0 ••• 0 0 0 0 1 0 ••• 0 0 0 0 0 1 ••• 0 0 0 . . . . . . . . . . . . . . . . . . . . . 0 0 0 ••• 1 0 0 0 0 0 ••• 0 1 0     ,
where we have used the relation a n 1,1 = ( n i=1 a i,1 ) n • n i=2 (a n i,1 ) -1 . It follows from this description that ρ is injective, and from Lemma 5 and (3.3) that G n,g is a crystallographic group of dimension 2ng and whose holonomy group is Z n . Now we prove that G n,g is torsion free. Let ω ∈ G n,g be an element of finite order. Since L is torsion free, the order of ω is equal to that of σ(ω) in the cyclic group G n , in particular ω n = 1. Using (3.1) and (3.2), as well as the fact that L is torsion free, there exist θ ∈ L and j ∈ {0, 1, 2, . . . , n -1} such that ω = θ(a 1,1 α n-1 ) j . Making use of the basis

Y ′ of L, θ = n i=1 a i,1 λ 1,1 . n i=2 a nλ i,1 i,1 . 2g r=2 n i=1 a nλ i,r i,r , (3.5)
where λ i,r ∈ Z for all i = 1, . . . , n and r = 1, . . . , 2g. On the other hand:

1 = ω n = n-1 k=0 (a 1,1 α n-1 ) jk θ(a 1,1 α n-1 ) -jk (a 1,1 α n-1 ) nj . (3.6)
By (3.2), (a 1,1 α n-1 ) nj = n i=1 a j i,1 , and thus the right-hand side of (3.6) belongs to P n (M)/Γ 2 (P n (M)). We now compute the coefficient of a 1,1 in (3.6) considered as one of the elements of the basis of P n (M)/Γ 2 (P n (M)) given by Corollary 7. From (3.1), the terms appearing in the product 2g r=2 n i=1 a nλ i,r i,r of (3.5) do not contribute to this coefficient. Since conjugation by a 1,1 α n-1 permutes cyclically the elements a 1,1 , a 2,1 , . . . , a n,1 by (3.1), it follows that conjugation by a 1,1 α n-1 leaves the first term

n i=1 a λ 1,1
i,1 of (3.5) invariant, and so with respect to (3.6), it contributes nλ 1,1 to the coefficient of a 1,1 . In a similar manner, with respect to (3.6), the second term of (3.5) contributes n(λ 2,1 + • • • + λ n,1 ) to the coefficient of a 1,1 . Putting together all of this information, the computation of the coefficient of a 1,1 in (3.6) yields the relation n(λ 1,1 + λ 2,1 + • • • + λ n,1 ) + j = 0. We conclude that j = 0, so ω = θ = 1 because L is torsion free, which using the first part of the statement, shows that G n,g is a Bieberbach group.

To prove the last part of the statement, using [START_REF] Szczepański | Geometry of crystallographic groups[END_REF]Lemma 5.2(3)] and the fact that G n is cyclic, the centre Z( G n,g ) of G n,g is given by:

Z( G n,g ) = {θ ∈ L | ρ(g)(θ) = θ for all g ∈ G n } = {θ ∈ L | ρ((1, n, . . . , 2))(θ) = θ}. (3.7)
To compute Z( G n,g ), let θ ∈ L. Writing θ with respect to the basis Y ′ of L as a vector

β 1 . . . β 2g
, where for all i = 1, . . . , 2g, β i is a column vector with n elements, and using the description of the action ρ given by (3.4), it follows that θ ∈ Z( G n,g ) if and only if β i belongs to the eigenspace of M i with respect to the eigenvalue 1 for all i = 1, . . . , 2g. It is straightforward to see that these eigenspaces are of dimension 1, and are generated by if i = 2, . . . , 2g. We conclude using (3.7) that Z( G n,g ) is the free Abelian group generated by { n i=1 a i,1 , n i=1 a n i,r | 2 ≤ r ≤ 2g}. This generating set may be seen to be a basis of Z( G n,g ), in particular, Z( G n,g ) is free Abelian of rank 2g.

We do not know whether B n (M)/Γ 2 (P n (M)) admits a Bieberbach subgroup of maximal rank whose holonomy group is non Abelian. The following proposition shows that a certain Frobenius group cannot be the holonomy of any Bieberbach subgroup of B n (M)/Γ 2 (P n (M)).

Proposition 13. Let p be an odd prime, and let M be a compact, orientable surface without boundary of genus g ≥ 1. In B p (M)/Γ 2 (P p (M)) there is no Bieberbach subgroup H such that σ(H) is isomorphic to the Frobenius group Z p ⋊ θ Z (p-1)/2 , where the automorphism θ(ι (p-1)/2 ) is of order (p -1)/2, ι (p-1)/2 being a generator of Z (p-1)/2 .

Proof. Let H be a subgroup of B p (M)/Γ 2 (P p (M)) such that σ(H) is isomorphic to the Frobenius group Z p ⋊ θ Z (p-1)/2 . Let us show that H has non-trivial elements of finite order. Using Proposition 1, we also identify B p (M)/Γ 2 (P p (M)) with Z 2gp ⋊ S p . Each element B p (M)/Γ 2 (P p (M)) may thus be written as x. ψ(w) where x ∈ P p (M)/Γ 2 (P p (M)) and w ∈ S p , which we refer to as its normal form. As in the proofs of Theorem 2(b) and Proposition 12, P p (M)/Γ 2 (P p (M)) splits as a direct sum ⊕ 2g 1 Z p that we interpret as a Z[S p ]-module, the module structure being given by Proposition 8. If z ∈ P p (M)/Γ 2 (P p (M)) then for j = 1, . . . , 2g, let z j denote its projection onto the jth summand of ⊕ 2g 1 Z p , and for (z, τ ) ∈ Z 2gp ⋊ S p , let (z, τ ) j = (z j , τ ) ∈ Z p ⋊ S p . Let ε : Z p -→ Z denote the augmentation homomorphism. We extend ε to a map from Z p ⋊ S p to Z, also denoted by ε, by setting ε(z, τ ) = ε(z) for all (z, τ ) ∈ Z p ⋊ S p . Using the Z[S p ]-module structure, observe that: ε((λzλ -1 ) j ) = ε(z j ) for all λ ∈ B p (M)/Γ 2 (P p (M)) and z ∈ P p (M)/Γ 2 (P p (M)).

(3.8)

Hence for all (z, τ ), (z ′ , τ ′ ) ∈ Z p ⋊ S p :

ε(zτ. z ′ τ ′ ) = ε(zτ z ′ τ -1 . τ τ ′ ) = ε(z. τ z ′ τ -1 ) = ε(z). ε(τ z ′ τ -1 ) = ε(z) + ε(z ′ ) = ε(z, τ ) + ε(z ′ , τ ′ ),
and thus ε :

Z p ⋊ S p -→ Z
v 2 v 1 v -1 2 v -l 1 ∈ H ∩ P p (M)/Γ 2 (P p (M)). Further: v 2 v 1 v -1 2 v -l 1 =a 2 ψ(w 2 )a 1 ψ(w 1 )ψ(w 2 ) -1 a -1 2 (a 1 ψ(w 1 )) -l =a 2 ψ(w 2 )a 1 ψ(w 2 ) -1 a -1 2 . a 2 . ψ(w 2 w 1 w -1 2 )a -1 2 ψ(w 2 w 1 w -1 2 ) -1 . ψ(w 2 w 1 w -1 2 w -l 1 ). l k=1 (ψ(w 1 ) l-k a -1 1 ψ(w 1 ) k-l ),
and applying (3.8) and using the relation

w 2 w 1 w -1 2 = w l 1 , it follows that: ε((v 2 v 1 v -1 2 v -l 1 ) i ) = (1 -l)ε((a 1 ) i ) (3.9) for all 1 ≤ i ≤ 2g. Let v = v 2 v 1 v -1 2 v -l 1 v l-1 1 . The element v l-1 1 also belongs to H, so v ∈ H, and since v l-1 1 = (a 1 ψ(w 1 )) l-1 = l-2 k=0 (ψ(w 1 ) k a 1 ψ(w 1 ) -k ) ψ(w 1 ) l-1 , for all 1 ≤ i ≤ 2g, it follows that (v) i = β i ψ(w 1 ) l-1
, where β i ∈ Z p is given by:

β i = (v 2 v 1 v -1 2 v -l 1 ) i . l-2 k=0 (ψ(w 1 ) k a 1 ψ(w 1 ) -k ) i . (3.10)
Using (3.8), (3.9) and (3.10), we see that:

ε(β i ) = ε((v 2 v 1 v -1 2 v -l 1 ) i )+ε l-2 k=0 (ψ(w 1 ) k a 1 ψ(w 1 ) -k ) i = (1-l)ε((a 1 ) i )+(l-1)ε((a 1 ) i ) = 0 (3.11)
for all 1 ≤ i ≤ 2g. Now in normal form, v may be written v = (β 1 , . . . , β 2g )ψ(w 1 ) l-1 . Since w l-1 1 is non trivial, it follows that v is non trivial. Taking z = v and k = p in Lemma 11 and using (3.11) it follows that v is of order p, and hence H has non-trivial torsion elements. In particular, H is not a Bieberbach group.

It seems to be an interesting question to classify the subgroups of S n which can be the holonomy of a Bieberbach subgroup of B n (M)/Γ 2 (P n (M)) of maximal rank. In the case where the subgroup is a semi-direct product, the argument given in the proof of Proposition 13 may be helpful in the study of the problem.

Using the holonomy representation of the Bieberbach group G n,g of Theorem 3, given in equation (3.4), we now prove some dynamical and geometric properties of the flat manifold X n,g whose fundamental group is G n,g .

Proof of Theorem 4. Let n ≥ 2, let g ≥ 1, let X n,g be a Riemannian compact flat manifold X n,g whose fundamental group is G n,g , the Bieberbach group given in the statement of Theorem 3, and let G n be the cyclic group of that theorem. Let 1 denote the generator (1, n, n -1, . . . , 2) of G n , and consider the holonomy representation ρ : Z n -→ Aut(Z 2ng ) of G n,g given in the proof of Theorem 3. By (3.4), if the characteristic polynomial of ρ( 1) is equal to (x n -1) 2g , where ρ : Z n -→ Aut(Z 2ng ) is the holonomy representation of G n,g . To see this, if 2 ≤ i ≤ 2g then M i is the companion matrix of the polynomial x n -1, and if we remove the first row and column of M 1 , we obtain the companion matrix of the polynomial 1 + x + x 2 + • • • + x n-1 , so the characteristic polynomial of M 1 is also equal to (x -1)(1 + x + x 2 + • • • + x n-1 ) = x n -1. In particular, det(ρ(1)) = 1, from which it follows from the end of Section 2.1 that X n,g is orientable. Further, the eigenvalues of ρ(1) are the nth roots of unity each with multiplicity 2g, and we conclude from [START_REF] Porteous | Anosov diffeomorphisms of flat manifolds[END_REF]Theorem 7.1] that X n,g admits Anosov diffeomorphisms. By [5, Theorems 6.4.12 and 6.4.13], the first Betti number of X n,g is given by: β 1 (X n,g ) = 2ng -rank(ρ(1) -I 2ng ) = 2ng -2g(n -1) = 2g.

It remains to show that X n,g admits a Kähler structure. In order to do this, we make use of the following result from [15, Theorem 3.1 and Proposition 3.2] (see also [6, Theorem 1.1 and Proposition 1.2]) that a Bieberbach group Γ of dimension m is the fundamental group of a Kähler flat manifold with holonomy group H if and only if m is even, and each R-irreducible summand of the holonomy representation ψ : H -→ GL(m, R) of Γ, which is also C-irreducible, occurs with even multiplicity. Since dim(X n,g ) = 2ng and the character vector of the representation ρ is equal to 2ng 0 . . . 0 , it follows that each real irreducible representation of ρ appears 2g times in its decomposition, and hence that X n,g admits a Kähler structure.

The Betti numbers of the Kähler manifold X n,g may be computed using the formula β i (X n,g ) = dim(Λ i (R 2n )) Gn of [6, Page 370]. For real dimensions 4 and 6, we may identify the fundamental group of X n,g in the CARAT classification [START_REF] Opgenorth | Carat, Crystallographic AlgoRithms And Tables[END_REF] and [START_REF] Dekimpe | Kähler flat manifolds[END_REF]Tables,pp. 368 and 370]. More precisely, the CARAT symbol of the 4-dimensional Bieberbach group G 2,1 (resp. the 6-dimensional Bieberbach group G 3,1 ) is 18.1 (resp. 291.1).

The cases of the sphere and non-orientable surfaces without boundary

Let M be a compact, connected surface without boundary. In this section, we describe the quotient group B n (M)/Γ 2 (P n (M)) for the cases not covered by Proposition 1, namely M is either the 2-sphere S 2 or a compact, non-orientable surface of genus g ≥ 1 without boundary, which we denote by N g . If g = 1 then N 1 is the projective plane RP 2 . Lemma 14. Let M be either the 2-sphere S 2 or a compact, non-orientable surface of genus g ≥ 1 without boundary, and let n ∈ N. The group P n (M)/Γ 2 (P n (M)) is isomorphic to: (a) Z 2 ⊕ Z n(n-3)/2 if M = S 2 and n ≥ 3. Further, the Γ 2 (P n (M))-coset of the full twist ∆ 2 n is the generator of the summand Z 2 . (b) Z n 2 ⊕ Z (g-1)n if M = N g . Proof. Part (a) follows from [8, page 674]. For part (b), if g = 1 then M = RP 2 , and the result is a consequence of [START_REF] Gonçalves | Inclusion of configuration spaces in Cartesian products, and the virtual cohomological dimension of the braid groups of S 2 and RP 2[END_REF]Proposition 8]. So suppose that g ≥ 2. We make use of [13, Theorem 5.1, Presentation 3]. Since T i,i = 1 for all 1 ≤ i ≤ n [13, page 439] then by relation (Pr3) of that presentation, T i,j = T i,j-1 in P n (M)/Γ 2 (P n (M)) for all 1 ≤ i < j ≤ n, and it follows by induction on j -i that T i,j = 1. We conclude from the presentation that P n (M)/Γ 2 (P n (M)) is generated by

Theorem 2 .

 2 Let n ≥ 2, and let M be an orientable surface of genus g ≥ 1 without boundary. (a) Let e 1 and e 2 be finite-order elements of B n (M)/Γ 2 (P n (M)). Then e 1 and e 2 are conjugate if and only if their permutations σ(e 1 ) and σ(e 2 ) have the same cycle type. Thus two finite cyclic subgroups H 1 and H 2 of B n (M)/Γ 2 (P n (M)) are conjugate if and only if the generators of σ(H 1 ) and σ(H 2 ) have the same cycle type. (b) If H 1 and H 2 are subgroups of B n (M)/Γ 2 (P n (M)) that are isomorphic to S n then they are conjugate.

  is a consequence of relations (R7) and (R8) of [13, Theorem 4.2, step 3], with the exception of the case j = i, i + 1, which is clear. Part (b) is relation (R9) of [13, Theorem 4.2, step 3], and part (c) is relation (PR3) of [13, Theorem 4.2, presentation 1]

  be as in the statement, let σ(z) = (l 1 , . . . , l m ), where m ≥ 2, and let ω = ψ(σ(z)). By Proposition 1, ω is of order m, and the decomposition z = ω

1

 1

  2.8). Thus θ is conjugate to ω, which proves the claim, and thus the first part of the statement. (b) We start by showing that if H ⊂ B n (M)/Γ 2 (P n (M)) is isomorphic to S n then H and ψ(σ(H)) are conjugate. This being the case, it follows that each of the subgroups H 1 , H 2 is conjugate to ψ(σ(H)), and the result follows. Suppose that H is a group isomorphic to S n that embeds in (Z

n ⊕ • • • ⊕ Z n ) ⋊ S n .

Using the Z[S n ]-module structure of Z 2ng given above, it follows that H embeds in Z n ⋊ S n , for any one of 2g summands of Z n . Let us first prove that the result holds for such an embedding. Let s : H -→ Z n ⋊ S n be an embedding, let π : Z n ⋊ S n -→ S n be projection onto the second factor, and let ψ : S n -→ Z n ⋊ S n be inclusion into the second factor.

  is a homomorphism. Identifying σ(H) with the Frobenius group Z p ⋊ θ Z (p-1)/2 , let w 1 , w 2 ∈ S p be generators of Z p and Z (p-1)/2 respectively. For i = 1, 2, let v i ∈ H be such that σ(v i ) = w i , and let a i ∈ P p (M)/Γ 2 (P p (M)) be such that v i = a i ψ(w i ), where ψ : S p -→ B p (M)/Γ 2 (P p (M)) is the section for σ given in the proof of Proposition 1. Using the relation w 2 w 1 w -1 2 = w l 1 in the Frobenius group, where l is an element of the multiplicative group Z *

p of order (p -1)/2, we have

{a j,r | 1 ≤ j ≤ n and 1 ≤ r ≤ g}, subject to the relations a 2 j,1 a 2 j,2 • • • a 2 j,g = 1 for all 1 ≤ j ≤ n, from which we obtain the given isomorphism.

Proposition 15. Let M = N g , where g ≥ 1. In terms of the presentation of B n (M) given by [13, Theorem 2.2], the map ψ : S n -→ B n (M)/Γ 2 (P n (M)) defined on the generating set {τ i | i = 1, . . . , n-1} of S n by ψ(τ i ) = σ i for all 1 ≤ i ≤ n -1 is an injective homomorphism. Consequently, the short exact sequence (1.2) splits, and

) ⋊ S n Proof. Since τ 1 , . . . , τ n-1 and σ 1 , . . . , σ n-1 satisfy the Artin relations, and using (1.2), it suffices to show that

) for all i = 1, . . . , n -1. We now prove that this is the case. If g = 1 (resp. g ≥ 2), with the notation of [9, Theorem 7] (resp. [13, Theorem 2.2]),

) in P n (M) for all 1 ≤ i ≤ n -1, so σ 2 i belongs to Γ 2 (P n (M)) by [9, Proposition 8] (resp. by the proof of Lemma 14), and thus its Γ 2 (P n (M))-coset is trivial in B n (M)/Γ 2 (P n (M)) as required.

By Lemma 14, if M = S 2 or N g , where g ≥ 1, P n (M)/Γ 2 (P n (M)) has torsion, and we cannot use the methods of the proof of Proposition 1 . But in fact, in these cases, the group B n (M)/Γ 2 (P n (M)) is not crystallographic. To see this, we first prove the following lemma.

Lemma 16. Let Π be a group, and suppose that there exists a group extension of the form:

where H is torsion free, and T is finite and non-trivial. Then Π is not a crystallographic group.

Proof. To prove that Π is not crystallographic, by [5, page 34], it suffices to show that it possesses a non-trivial, normal finite subgroup. Let us show that T is such a subgroup. By the hypotheses, it suffices to prove that T is normal. To see this, we view the kernel T × H as a subgroup of Π. Any inner automorphism of Π thus restricts to an automorphism of T × H. Since the image by such an automorphism of any element of T is also a torsion element, it follows from the fact that H is torsion free that T is indeed normal in Π.

Proposition 17. Let M = S 2 or N g , where g ≥ 1. Then for all n ≥ 1, the quotient B n (M)/Γ 2 (P n (M)) is not a crystallographic group.

Proof. The result follows from (1.2) and Lemmas 14 and 16.

Remark 18. If M = S 2 (resp. M = N g , where g ≥ 2), the subgroup T is that generated by the class of the full twist braid (resp. by {a j,1 a j,2 • • • a j,g | j = 1, . . . , n} using the notation of [START_REF] González-Meneses | New presentations of surface braid groups[END_REF]Theorem 5.1]). If M = RP 2 then T = P n (RP 2 )/Γ 2 (P n (RP 2 )).


Sato Masayuki
email: sato.masayuki@jaxa.jp

Georges Hardier

Gilles Ferreres

Christopher Edwards

Halim Alwi

Lejun Chen

Andres Marcos

Masayuki Sato

* ・ Georges

Hardier **

・ Gilles

Ferreres **

*** ・ Halim

Alwi ***

・ Lejun

Chen ***

Andres Marcos ****

Fault Tolerance -Background and Recent Trends

Keywords: Words:fault tolerant control, fault detection and identification, adaptive control, sliding mode control/observer, H∞ control

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

In 1985, a Boeing 747 airplane crashed on Mt. Osutaka with over 500 persons dead. Japan Transport Safety Board investigated the accident, and concluded that the power loss of the main control devices, which was caused by the break of the aft pressure bulkhead, led to this fatal accident 1) . This accident illustrates the importance of fault tolerant flight control.

In this article, several schemes related to the field of Fault Detection and Isolation (FDI), Fault Detection and Diagnosis (FDD), and Fault Tolerant Control (FTC) are presented both from a theoretical perspective and, more importantly, from an application perspective. The focus of the applications is the aeronautical domain and they cover a range of projects undertaken during the last two decades. These projects (some of them funded by the European Commission) showcase the time-line followed in Europe for the evolution and transfer of aircraft Fault Detection, Identification and Reconfiguration (FDIR) scheme, which include FDI/FDD/FTC modules, from academia to the industrial sector.

Three main approaches are presented: (i) Adaptive control (Section 2), (ii) Sliding Mode Control/Observers (SMC/SMO) (Section 3), and (iii) H ∞ control (Section 4).

FDD/FTC via indirect adaptive control

It is well established that Fly-By-Wire (FBW) improves the safety of aircraft significantly. However, the implementation of advanced capabilities for detecting, protecting and optimizing guidance and control [START_REF]便の御巣鷹山墜落事故に係る航空事故調査報告書に ついての解説:運輸安全委員会[END_REF] requires the availability of aircraft (A/C) states, especially for critical parameters such as the Angle of Attack (AoA) and the Calibrated Air Speed (CAS). Hence, a major issue for the implementation of these FBW advances is their capability to cope with faulty situations (of the corresponding sensors) in order to avoid switching to degraded control modes (i.e. to reduced control laws and flight envelope protections). This capability can only be achieved using specific FDD and FTC schemes against these types of faults.

This section focuses on the augmentation of control laws' availability to provide the Flight Control System (FCS) with the ability to automatically accommodate potential failures or unexpected events. To fill the gap between academic FDD/FTC and industrial practice 2), [START_REF] Zolghadri | Advanced model-based FDIR techniques for aerospace systems: Today challenges and opportunities[END_REF] , it is always required to use highfidelity, nonlinear simulators as well as industrial implementation constraints that address realistic, onboard issues 4), [START_REF] Kerr | RECONFIGURE FP7 project: preliminary results and contributions[END_REF] . Specifically, when some key parameters are missing (e.g., the airspeed), the approaches described in this section consist in scheduling the control laws according to estimated versions of those parameters. This must be achieved without degrading the performance and is based on "advanced flight control laws" (see the corresponding block in Fig. 1). Two possible alternatives are examined in here: to replace the faulty measurement by an estimated value (virtual sensing) while keeping the same scheduling strategy (subsection 2.1), or to schedule the laws based on another type of parameter, namely some model coefficients (subsection 2.2). In both cases, the presented approaches rely on so-called Linear Fractional Transformation (LFT) forms (subsection 2.2), and results in analytical scheduling expressions that facilitate the use of robust control design and analysis tools. A more advanced solution relies on analytical redundancy, which refers to the use of existing dynamical relationships between A/C states. This solution relies on sensor data supplemented by a model-based simulation of the flight mechanics to deliver the estimation of the required parameters (the "state estimation" block of Fig. 1). These estimates can be processed in parallel with the measured signals contributing to extend the way the flight parameters are consolidated (the "advanced FDD" block in Fig. 1). The concept of analytical redundancy for FCS is not new [START_REF] Goupil | Oscillatory failure case detection in the A380 electrical FCS by analytical redundancy[END_REF] while that of virtual sensors is more recent.

ONERA has been working with Airbus on this topic for a decade and has developed adaptive versions of several nonlinear state estimation techniques to deliver AoA and CAS estimates in faulty conditions [START_REF] Hardier | Model-based techniques for virtual sensing of longitudinal flight parameters[END_REF] . A major obstacle to an operational use of this kind of approaches springs from their implementation in real time (especially from the computational burden of the algorithms) but also from the certification process (which impose stringent requirements to be fulfilled). Several solutions have been proposed by ONERA to limit the complexity of the developed Adaptive Extended Kalman Filter (AEKF): for example, the use of kinematic relationships and random walk processes [START_REF] Alcalay | Development of virtual sensors to estimate critical aircraft flight parameters[END_REF] for the process equations, as well as introducing a priori knowledge only in one output equation (vertical load factor) by means of a simplified modeling of the lift aerodynamic coefficient. The latter approximation is achieved by a set of surrogate models that facilitate on-board implementation and signal differentiation. More details are given in 9) in regards to these surrogate models and the principles of the techniques that permit to obtain rationaltype representations (using the APRICOT library of the SMAC Toolbox, w3.onera.fr/smac/?q=apricot, developed by ONERA). As the FDD module is a component of the AEKF 7), 8) , the gain matrix calculation can be adapted continuously, in case of detected/isolated faults, by simply omitting their values in the sequential update process. The tedious linearizations are also replaced by analytical derivations (for the process equations that only involve kinematic expressions as well as for those outputs that rely on surrogate models 9) ).

This monitoring/estimation process called FPE was evaluated in RECONFIGURE European project by means of Monte-Carlo campaigns (using the highfidelity simulator provided by Airbus as well as realistic flight conditions reproducing in-service and worstcase extreme events). Patents are currently being registered by Airbus and ONERA (see 10) for a preliminary version), and advanced validation work is on-going 8), [START_REF] Alcalay | An adaptive Extended Kalman Filter for monitoring and estimating key aircraft flight parameters[END_REF] .

It is noted that by using the AoA and/or CAS consolidated estimates as scheduling parameters (for protection or control laws) either the nominal control architecture can be kept or a new gain-scheduled controller can be designed based on the estimated signals. Their principles are given in the next subsection.

Indirect Adaptive Control

Gain-scheduling is classically used by industry to control an airplane throughout its flight envelope. When a scheduling parameter is unavailable due to a sensor fault, adaptive control is an attractive solution to recover closed loop performance by scheduling the gains with respect to, for example, the aerodynamic model coefficients instead of the usual flight parameters. The principle is to estimate some of the model parameters (identification block in Fig. 1) and to adjust the controller on-line using the updated estimates.

For the gain-scheduling control design component, several state-space techniques are available based on the use of time-varying or time-invariant scheduling parameters. In the approach proposed in 12), the modal technique of 13) is used as it appears especially suitable for A/C applications. It requires building first an open loop LFT model using ONERA's LFR Toolbox (w3.onera.fr/smac/?q=lfrt) using the main aerodynamic coefficients as model parameters (whose ranges of variation correspond to the set of linearized models used to design the controller). Then, a static output feedback placing the main closed loop poles is synthesized also in an LFT form. In 12), a surrogate model of the LFT gains was derived off-line to minimize the onboard computation (using also the SMAC/APRICOT Toolbox).

The second issue of the adaptation process requires the model parameters to be estimated on-line [START_REF] Hardier | A Recursive Estimation Algorithm to Track Aircraft Model Parameters[END_REF] with the capability to track time-varying coefficients in the face of measurement noise, poor data content, and external disturbances (turbulence). Residual errors will sometimes be large and a measure of the parameter accuracy should also be provided so that some logic can be introduced: e.g., if the estimated parameters are not reliable enough, one can switch to a robust back-up controller ensuring minimal performance properties [START_REF] Hardier | A Recursive Estimation Algorithm to Track Aircraft Model Parameters[END_REF] . To address this issue, algorithms for recursive parameter estimation are used, for example: Time Domain (TD) methods are based typically on recursive least-squares, or on Extended Kalman Filter (EKF) when coping with non-linearities. However, onboard computer limitations preclude the use of these methods, e.g. any iterative algorithm resulting in some unbounded computation loop through the data. Hence complex operations like non-analytical matrix inversions should be avoided. Accordingly, TD algorithms seem preferable as the most basic versions only involve very simple operations (matrix inversion replaced by a scalar division) [START_REF] Hardier | An extended U-D algorithm with multiple forgetting factors for RLS estimation of model parameters[END_REF] . Another solution comes from the Frequency Domain (FD) techniques which have many desirable features [START_REF] Hardier | A Recursive Estimation Algorithm to Track Aircraft Model Parameters[END_REF] : e.g. computation time is reduced by processing only a limited amount of frequencies within the bandwidth of interest, and the resulting indirect filtering of the wide-band disturbances (e.g., low frequency mismatch or high frequency noise) improves estimation accuracy 16), 17) .

Otherwise, TD and FD estimation schemes can rely on a fully recursive algorithm (i.e. using measurements as soon as they are available) or alternatively on a kind of sequential procedure, processing m o v i n gd a t aw i n d o w sw i t hal o w e rr a t et og e tas u ccession of piecewise constant values. Such an approach was used for actuator faults' FDD during the ONERA/DLR research project IMMUNE 17), 18) .I n 12), 14), it is shown that FD approaches can also be simplified to reduce their complexity, similarly to what is done for TD formulations, and they can also include similar advanced mechanisms such as data forgetting 15), [START_REF] Ljung | Theory and practice of recursive identification[END_REF] . It can also be used for FDD purposes by monitoring the changes in the parameters to diagnose unexpected behaviors [START_REF] Hardier | On-line parameter identification for in-flight aircraft monitoring[END_REF] .

The described indirect adaptive process has been evaluated also within the RECONFIGURE project using the high-fidelity simulator provided by Airbus 12), [START_REF] Hardier | A Recursive Estimation Algorithm to Track Aircraft Model Parameters[END_REF] . It is also noteworthy that the worst-case validation of a similar adaptive scheme was already investigated by Airbus and ONERA [START_REF] Oudin | Design and worst-case validation of a longitudinal adaptive flight control law: a practical approach[END_REF] .

FTC and FDI via Sliding Mode Scheme

SMC schemes have a number of interesting properties, which fuelled the continued research interest in this field since the 1960s. Perhaps the key property is its inherent insensitivity (at least theoretically) to so-called matched uncertainty -i.e. uncertainty acting in the channel of the input control signals 21), 22) . Many different paradigms for the design of linear sliding surfaces for uncertain linear systems have been developed, and this area of research is quite mature 21), 22) . In conventional sliding modes the closedloop behaviour can be split into two well-defined distinguishable phases: a) the pre-sliding phase in which the controller drives the system states towards the sliding surface prior to achieving a sliding mode; and b) the reduced order sliding motion which occurs once the surface is attained and the states are forced to evolve along this surface. Since actuator faults can be considered as matched uncertainty, sliding mode control is a good candidate paradigm for FTC. The combination of SMC with Control Allocation 23) (thought of as a mechanism for distributing control signals in over-actuated systems) has the benefit of providing SMCs with access to redundant actuators, and thus enabling the overall control scheme to handle actuator failures in addition to faults.

FTC via Sliding Mode Control

Figure 2 illustrates the overall FTC strategy [START_REF] Alwi | Fault tolerant control using sliding modes with on-line control allocation[END_REF] where it is assumed that the Control Application (CA) module depends on the effectiveness of the actuators. The information necessary to compute these quantities on-line can be supplied by a FDI/FDD scheme, or by using a measurement of the actual actuator deflection compared to the demanded deflection. If an actuator fault occurs, the actuator effectiveness (written as W in Fig. 2) will be changed online and the control input is reallocated to minimize the use of the faulty surfaces [START_REF] Alwi | Fault tolerant control using sliding modes with on-line control allocation[END_REF] .

Fig. 2 Control allocation strategy

To ensure robustness and performance over a wide range of operating conditions, the original sliding mode control allocation scheme presented in 24) has been extended within a Linear Parameter Varying (LPV) framework [START_REF] Alwi | Real-time implementation of an ISM fault-tolerant control scheme for LPV plants[END_REF] . This FTC-SMC scheme has been implemented on various industrial applications. For example, during the GARTEUR FM-AG16 project [START_REF]Fault Tolerant Flight Control: A Benchmark Challenge[END_REF] it was applied to the ADMIRE and the RECOVER benchmark aircraft models (to cope with elevator and stabilizer failures) and then tested on the Technical University of Delft SIMONA piloted motion simulator 25), 26) (Fig. 3).

Continuing the above research, during the RE-CONFIGURE project [START_REF] Alwi | Fault Detection and Fault Tolerant Control Using Sliding Modes[END_REF] , an Integral Sliding Mode (ISM) approach was adopted 22) to deal with random elevator failures. The resulting industrial evaluation of the scheme showed a very low computational load indicating no limitations for future onboard implementation. The main advantage of the scheme, as shown in Fig. 4, is that it can be retro-fitted to existing control loops without the need for their redesign.

FDI via Sliding Mode Observers

Sliding mode ideas can also be considered in an observer context. The novelty of SMO is their ability to reconstruct un-measurable signals within a process by appropriate scaling and filtering of the socalled "equivalent output error injection" 22) .T h i si s a unique property of SMOs which emanates from the fact that the introduction of a sliding motion forces the outputs of the observer to perfectly track the plant measurements.

As part of the RECONFIGURE project, the problem of estimating fault signals using SMOs was considered based on an additive perturbation model of the form ẋ(t)=Ax(t)+Bu(t)+Ff i (t, u)+Mξ(t, y, u) where f i represents actuator faults and ξ encapsulates uncertainty. The aim was to design an SMO of the form

ż(t)=Az(t)+Bu(t) -G l e y (t)+G n v(t)
where e y = C(zx)a n dv(t)=-ρ o (t, y, u)sign(e y ), to force and maintain sliding on S = {e : Ce =0 } with e = z-x representing the state estimation error. A fixed gain H ∈ R q×p is chosen to form a reconstruction signal fi = Hv eq ,w h e r ev eq is the equivalent injection signal. This is quite different from traditional linear residual based methods since even in the presence of faults the output estimation remains zero.

The reconstruction signal is designed by choice of H, G n , so that: (i) it tends to f i when ξ =0(atw orst asymptotically), and (ii) the L 2 gain γ between ξ and the fault residual is minimised (a convex problem).

The ADDSAFE benchmark included a scenario with a potentially faulty yaw rate signal emerging from the air data and inertial reference system. A sliding mode scheme based on the observer structure above was "flight-certified" coded for this scenario and validated on the Airbus Flight Control Computer (FCC) demonstrating its FDI performance and robustness via Monte Carlo campaigns [START_REF] Alwi | Development and application of sliding mode LPV fault reconstruction schemes for the ADDSAFE Benchmark[END_REF] .

FDD and FTC via H ∞ Control

FDD and FTC approaches based on H ∞ optimization have been proposed and consolidated in academia for the last 20 years 29), [START_REF] Marzat | Model-based fault diagnosis for aerospace systems: a survey[END_REF] . Very relevant success stories of H ∞ controllers flying (or [START_REF] Philippe | H∞ Control for Telecommunications Satellites, The Impact of Control Technology[END_REF] , the Ariane launcher [START_REF] Mauffrey | H∞ method applied to launcher control during atmospheric flight[END_REF] , and spacecraft such as the European Automated Transfer Vehicle (ATV) [START_REF] Bourdon | Position Control Design and Validation Applied to ATV During Docking to ISS[END_REF] ,E S A Rosetta [START_REF] Falcoz | Systematic design methods of robust and structured controllers for satellites[END_REF] , and CNES MICROSCOPE [START_REF] Pittet | Structured Accelero-Stellar Estimator for Microscope Drag-Free Mission, 3rd CEAS Specialist Conference on GNC[END_REF] .I n t h i s section a brief review of FDI/FTC H ∞ theory and its use on aerospace projects is presented.

Theory of Model-Based Approach

The essence of model-based FDI/FDD problem, on which H ∞ theory is based, is depicted in Fig. 5 and can be formalized as follows (assuming no uncertainty or disturbances besides faults): FDI/FDD problem: Given a model of a nominal system Gu and knowledge (measured or estimated) of the inputs u and outputs y of the system, the FDI/FDD problem consists of finding a filter F =[ F u F y ] T that provides a fault estimate res containing information on the actual faults f entering the system (through G f ).

The fault estimate (also known as residual) res can provide just an indication of the fault's presence (fault detection) or also the fault location and source (fault diagnosis) 29), 30), 36), [START_REF] Edelmayer | An H∞ filtering approach to robust detection of failures in dynamical systems[END_REF] . Fault detection requires only a single (scalar) value while fault diagnosis requires a set (vector) of fault estimates to distinguish between different faults. The residual definition is given by:

res = F y y + F u u = F y (G f f + G u u)+F u u = F y G f f +(F u + F y G u )u
It is now straight forward to identify the main objectives of an FDI/FDD scheme: i) Maximize the fault effects (using F y and the knowledge of the fault effects on the system G f ), and ii) minimize any other effect (e.g. controller input u, uncertainty, disturbances, other subsystem faults, etc.).

By adding the disturbance d effects to the output of the system y, the equation above is transformed into:

res = F y G f f + F y G d d +(F u + F y G u )u
This clearly shows that if a fault affects the system along the same direction as a disturbance (i.e. G f = G d ) then it is not possible to detect such fault. The above discussion is based on an openloop FDI/FDD design perspective, and although it is also possible to apply them to closed-loop systems, open-loop is typically favoured [START_REF] Niemann | Robust Fault Detection in Open Loop vs[END_REF] .

H ∞ FDIR Optimization

The problem of designing a robust FDI/FDD scheme is generally divided in to two main stages: robust residual generation and robust residual evaluation (if reconfiguration is also desired there is a third step that uses the latter's signals to compensate fault effects [START_REF] Zhang | Bibliographical review on reconfigurable fault-tolerant control systems[END_REF] ). Within the task of residual generation, and from the H ∞ perspective, the idea is to obtain a filter F =[ F u F y ] T so that: i) F u cancels the effects of the known inputs u, and ii) F y maximizes the effects of the faults f (while minimizing the effects of the disturbances d based on the frequency regions they act upon). This is accomplished using so-called weighting filters W ♯ which embed this frequency content knowledge as well as the desired design objectives into the H ∞ optimization framework. A general methodology is given in 36), 40)-42).

In order to exemplify the methodology, assume a nominal LTI plant G =[ G f G d G u ] T is given and it is desired to design an FDD filter F =[ F u F y ] T following a design rationale based on fault modelmatching (this strategy has several advantages for FDD, see 29) and 42)). First, the H ∞ FDI interconnection, see Fig. 6, is set then in this case three weights are defined to drive the optimization: W f (shaping the ideal fault knowledge and/or desired filter behaviour), W d (shaping the disturbances) and W u (for shaping the control command). The optimization will minimize the induced L 2 -norm of the transfer function from the inputs to the error. The shaping filters are used to emphasize the frequency regions where it is desirable to achieve these objectives, resulting in a direct way to perform the design trade-off.

Aerospace applications of H ∞ FDI/FTC

Similar to the previous Sections 2 and 3, the H ∞ approach has been matured from an aircraft FDIR Fig. 6 Ideal fault-matching FDI H ∞ interconnection perspective during the past 20 years. An incomplete list of projects and applications is given here, see the detailed survey of 30) for a more broad presentation.

( 1 ) Boeing 747 aircraft simulation model

The work of 42) is considered as the first instance that an H ∞ FDI filter was designed for a high-fidelity, full nonlinear aircraft simulation model [START_REF] Marzat | Model-based fault diagnosis for aerospace systems: a survey[END_REF] . The nonlinear time results (including sensor noise and gust) showed the good properties of the H ∞ filters despite aircraft motion changes arising from longitudinal maneuvers. This demonstration led to an active European-based impetus on the FDI/FTC topic starting with the establishment in 2004 of the GAR-TEUR FM-AG16 "Fault Tolerant Control" which ran until 2008 and included verification of a number of FDI approaches (see also previous sections) in a nonlinear simulator based on that from 43) and up to pilot simulations in SIMONA [START_REF] Alwi | Development and application of sliding mode LPV fault reconstruction schemes for the ADDSAFE Benchmark[END_REF] .

( 2 ) Piloted simulations in DLR ATTAS aircraft

As part of the German Aerospace Center (DLR) "Saturation Alleviation In-Flight Experiment" (SAIFE) test campaigns [START_REF] Brieger | Flight testing of low-order anti-windup compensators for improved handling and PIO suppression[END_REF] an H ∞ FDI scheme following that of 42) was demonstrated in piloted simulations. The SAIFE campaigns were performed on DLR Advanced Technologies Testing Aircraft (ATTAS), and focused on assessing the effectiveness of advanced anti-windup compensator against saturation [START_REF] Weston | Linear conditioning for systems containing saturating actuators[END_REF] . As these effects can be construed as unknown faults, FDI filters were designed using H ∞ and µ synthesis tools for the residual generation, and online thresholding algorithms for the residual evaluation [START_REF] Kerr | Piloted assessment of a fault diagnosis algorithm on the ATTAS aircraft[END_REF] . The results showed good FDI capabilities under quite demanding one-axis maneuvers.

( 3 ) Industrialization FDI: ADDSAFE EU project For the ADDSAFE project [START_REF] Goupil | The European ADDSAFE project: Industrial and academic efforts towards advanced fault diagnosis[END_REF] , and following the above two references 42), [START_REF] Kerr | Piloted assessment of a fault diagnosis algorithm on the ATTAS aircraft[END_REF] an H ∞ FDI filter was designed for detecting aileron actuator faults in the ADDSAFE Airbus benchmark. The specific scheme [START_REF] Marcos | Assessment on the ADDSAFE Benchmark Simulator of an H∞ Fault Detection Design for Aircraft[END_REF] followed a global aircraft H ∞ FDI scheme for the residual generation (i.e. using sensor measurements from the aircraft as opposed to only inputs/outputs of the actuator) and a simple time/magnitude logic for the residual evaluation (tasked with fault isolation between left and right inboard ailerons). The design was evaluated in a Monte Carlo campaign and was one of five (out of 13) qualified by Airbus as ready for industrial validation [START_REF] Goupil | Industrial benchmarking and evaluation of ADDSAFE FDD designs[END_REF] .

An additional outcome from ADDSAFE, but not related to H ∞ , was that one of the teams was able to further develop their design in collaboration with Airbus and recently it has been certified and implemented on the A350 family [START_REF] Zolghadri | Signal and model-based fault detection for aircraft systems[END_REF] for servo-loop oscillatory failures [START_REF] Goupil | Oscillatory failure case detection in the A380 electrical FCS by analytical redundancy[END_REF] . ( 4 ) Industrial FTC: RECONFIGURE EU project In reference 50), an FTC design based on the structured H ∞ synthesis approach (a more recent synthesis algorithm) was presented against elevon malfunctions for the RECONFIGURE benchmark. The structured H ∞ approach uses non-smooth optimization to obtain the gains of a pre-defined control architecture (thus preserving the legacy knowledge from industry) within an H ∞ formulation (thus allowing for a direct performance versus robustness tradeoff). The standard aircraft C * control architecture [START_REF] Field | The Application of a C * Flight Control Law to Large Civil Transport Aircraft[END_REF] was used for the FTC law, and its effectiveness was demonstrated using the Airbus benchmark [START_REF] Puyou | Tolerance of aircraft longitudinal control to the loss of scheduling information -towards a performance oriented approach[END_REF] .

Current Project

From March in 2016, a new Europe-Japan collaborative project called "VISION" (Validation of Integrated Safety-enhanced Intelligent flight cONtrol) has started. The main objective of this project is to increase Technology Readiness Level (TRL) for filling-up the gap between theories and practical systems on FTC/FDD schemes 3) . To this end, one of JAXA's research airplane MuPAL-α is used to demonstrate the usefulness and to examine the possible drawbacks of the methods above. In addition, Simple Adaptive Control (SAC) will be also implemented as a candidate of FTC control scheme.

Finally, all authors appreciate the funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No. 690811 and the Japan New Energy and Industrial Technology Development Organization under grant agreement No. 062800, as a part of the EU/Japan joint research project entitled "Validation of Integrated Safety-enhanced Intelligent flight cONtrol (VISION)." January 19, 2018)

(Received

Fig. 1

 1 Fig. 1 Schematic diagram of the proposed FDD/FTC blocks

Fig. 3

 3 Fig. 3 SIMONA flight simulator (Delft Univ. of Technology)

Fig. 5

 5 Fig.5Basic model-based FDI design architecture flown) are: telecommunication satellites[START_REF] Philippe | H∞ Control for Telecommunications Satellites, The Impact of Control Technology[END_REF] , the Ariane launcher[START_REF] Mauffrey | H∞ method applied to launcher control during atmospheric flight[END_REF] , and spacecraft such as the European Automated Transfer Vehicle (ATV)[START_REF] Bourdon | Position Control Design and Validation Applied to ATV During Docking to ISS[END_REF] ,E S A Rosetta[START_REF] Falcoz | Systematic design methods of robust and structured controllers for satellites[END_REF] , and CNES MICROSCOPE[START_REF] Pittet | Structured Accelero-Stellar Estimator for Microscope Drag-Free Mission, 3rd CEAS Specialist Conference on GNC[END_REF] .I n t h i s section a brief review of FDI/FTC H ∞ theory and its use on aerospace projects is presented.4.1 Theory of Model-Based ApproachThe essence of model-based FDI/FDD problem, on which H ∞ theory is based, is depicted in Fig.5and can be formalized as follows (assuming no uncertainty or disturbances besides faults):FDI/FDD problem: Given a model of a nominal system Gu and knowledge (measured or estimated) of the inputs u and outputs y of the system, the FDI/FDD problem consists of finding a filter F =[ F u F y ] T that provides a fault estimate res containing information on the actual faults f entering the system (through G f ).The fault estimate (also known as residual) res can provide just an indication of the fault's presence (fault detection) or also the fault location and source (fault diagnosis) 29), 30), 36),[START_REF] Edelmayer | An H∞ filtering approach to robust detection of failures in dynamical systems[END_REF] . Fault detection requires only a single (scalar) value while fault diagnosis requires a set (vector) of fault estimates to distinguish between different faults. The residual definition is given by: