Recension Of

Jodi Magness

Masada From

Michaël Girardin

Recension

Jodi Magness, Masada: from Jewish revolt to modern myth, Princeton -Oxford: Princeton University Press, 2019. Pp. 288. ISBN 9780691167107. $29,95. Michaël Girardin. Université du Littoral -Côte d'Opale. michael.girardin@univ-littoral.fr Preview As an Israeli national myth, the archaeological site of Masada remains one of the most visited in Israel and continues to exercise a deep impact on the imagination of thousands of people.

However, a great number of popular ideas about it have been proven to be false and the need for re-establishing the truth has been highly felt for a long time ago very sensible. Jodi Magness, one of the most famous archaeologists and specialists of Masada, succeeds in producing a book that is very accessible and up to date, a rewarding and stimulating work. The author's purpose is not only to propose a tour of Masada but to present the fortress in its broader context in geographical, historical, and socio-religious senses. The book is quite accessible even without background reading: one will find all the essentials about the Masada's background here.

The prologue (p. 1-4) offers long quotations of Flavius Josephus' War and raises the question of what we can know about these events, opening up the discussion by showing the contemporary importance of Masada for the Israeli State. Then begins a fascinating inquiry.

Chapter 1 presents the archaeological remains of the Roman siege works and the traces of their presence (notably military equipment). All suitable information about our main source of knowledge, Flavius Josephus, is then condensed in a few pages, with a specific attention to Josephus' biases and apologetic tendencies. <v>The five next chapters are devoted to describing the context of Masada.</v> Chapter 3, "Masada in context" explores Masada's natural setting, the Judean desert and the Dead Sea, then its "historical setting", listing the most important archaeological findings in the entire Judean desert since the Chalcolithic period up to the Byzantine period.

Chapter 4 presents Herod's works at the top of Masada as well as the other building projects of the king in Jerusalem, Caesarea Maritima, Samaria, Jericho and Herodium. Magness shows the patterns in Herodian architecture and makes it clear that Masada is only one of the Herodian palaces that could be interpreted in a broader background. The author provides, in this chapter, twenty three photographs that are illuminating and useful, some of them in colour.

Chapter 5 is dedicated to the history of Judea before Herod's kingship. It opens very early with the settling of Canaan around 1200 BCE and runs until the nomination of Herod as "king" at Rome in 40 BCE. The narrative is very short, but Magness takes care not to fall in the traps of many disputed events. She always avoids to take part in debates that are superfluous for her purpose.

Chapter 6 summarizes the history of Judea from 40 BCE to the uprising of 66 CE. Magness often refers to the New Testament in order to connect the story to the general culture of the reader, insisting in particular upon Pilatus, Felix and Festus among others Roman governors, and gives a balanced synthesis of the background of the Jewish revolt.

The revolt itself is the subject matter of chapter 7, with a focus first on its early developments and causes, a second one on Roman operations in the country and, third, on the siege of Jerusalem and the destruction of the temple. Magness pays attention to Josephus' description of the triumph, to the Flavian commemoration in Rome until Domitian's Principate and to the suppression of the revolt in Egypt as well as in Judea.

It is at this point that Magness's work shines the most. Chapter 8 surveys the rebel occupation of Masada, from 66 up to 74 CE. She beings by describing the state of current research. After having inquired about who the Jews at Masada were (probably a fluid population composed indeed of Sicarii but also of some Essenes), she shows the archaeological remains of the rebels' presence on the site, describing the housing, food, synagogue, the miqva'ot, stone and dung vessels that testify not only the ritual and purity concern of the rebels but also their poverty and low class origin. She surveys the sectarian scrolls that attest to a small Essene community here, then the clothing, the presence and role of women and, lastly, the rebels' administrative structure as reconstructed with the help of the seven hundred and one ostraca found on site. Twenty three iconographic documents are inserted here and help sustain the discussion.

The ninth and last chapter, entitled "Masada shall not fall again", explores the origins and developments of the Masada myth, with particular attention to the figure of Yigael Yadin, the first excavator of Masada and one of the most important promotors of its use in Zionist and nationalist discourses. Magness gives a short personal testimony about Yadin, who was one of her teachers at the Hebrew University of Jerusalem. She exposes the nationalist appropriation of the mass suicide of the Sicarii and, at the same time, the difficulties for archaeologists to confirm the story told by Josephus. In fact, it seems that no mass suicide was ever committed at Masada and that the story was only an embellishment of the narrative. But Magness advances here with a commendable prudence, not forgetting to conclude that she is not a Josephus specialist and that "whether or not the mass suicide story is true depends on how one evaluates Josephus's reliability as an historian". However, even if Masada does not play the same role in Israeli nationalism today, Magness shows that the site and its history still exercises a deep attraction on hearts and minds not only in Israel, but even around the world.

"Masada, she concludes, remains a symbol of the State of Israel and the Zionist enterprise".

In closing her presentation, Magness offers her readers "a tour of Masada" as an epilogue, i.e. her recommended itinerary for tourists and visitors. The notes, bibliography, a general index and image credits end the book.

In the chapters devoted to the context, one can sometimes feel that the author risks losing sight of the subject, because while trying to give her readers all suitable information, Magness spends a number of pages summarizing a lot of secondary data. In some ways, one could say that her Masada is constructed as a handbook on Jewish history at the turn of our era and, for specialists, the chapters about Masada proper seem to be too limited as a consequence. However, despite this small weakness, Masada remains a very beautiful book, well written and truly accessible for any interested reader. The book is up to date and presents in a fascinating manner the history of archaeological explorations in the area, the history of Zionism and of the State of Israel, and based on a few preceding works, of the ideological embellishment and misappropriations of the story of Masada. Undoubtedly, it will become a classic in this field of study.

  During the nineteenth century, explorations in the area of the Dead Sea led to finding the location of Masada. This "search for Masada" is summarized in chapter 2, with the (sometimes dramatic) adventures of Edward Robinson and Eli Smith, Christopher Costigan, William Molyneux, William Francis Lynch, Louis-Félicien de Saulcy, Henry Baker Tristram, and then with the project called the Survey of Western Palestine conducted by the Palestine Exploration Fund from 1871 to 1877. The archaeological explorations of Masada proper are listed with their most important contribution at the end of the chapter.

Princeton University Press, Princeton -Oxford, 2019. Bryn Mawr Classical Review, 2019. hal-


Stefan Friedl

Junghwan Park

Bram Petri

AND Jean Raimbault

Arunima Ray

On distinct finite covers of 3-manifolds

Every closed orientable surface S has the following property: any two connected finite covers of S of the same degree are homeomorphic (as spaces). In this, paper we give a complete classification of compact 3-manifolds with empty or toroidal boundary which have the above property. We also discuss related group-theoretic questions.

Introduction

Given a finitely generated group G and n ∈ N we denote by s n (G) the number of subgroups of index n and we denote by e n (G) the number of isomorphism types of subgroups of index n.

The study of s n (G) has attracted a lot of interest over the years. For example, the well-known asymptote for the number of index n subgroups in a free group F 2 of rank 2 is

s n (F 2 ) ∼ n • (n!) as n → ∞,
(see for instance [START_REF] Dixon | The probability of generating the symmetric group[END_REF] and [START_REF] Lubotzky | Subgroup growth[END_REF]Chapter 2]). Much more recently, similar asymptotes were determined for surface groups by Müller and Schlage-Puchta [START_REF] Müller | Character theory of symmetric groups and subgroup growth of surface groups[END_REF], Fuchsian groups by Liebeck and Shalev [START_REF] Liebeck | Fuchsian groups, coverings of Riemann surfaces, subgroup growth, random quotients and random walks[END_REF] and non-uniform lattices in higher rank simple Lie groups by Lubotzky and Nikolov [START_REF] Lubotzky | Subgroup growth of lattices in semisimple Lie groups[END_REF]. Finer questions ask for the number of subgroups of a given type. For instance, in [START_REF] Müller | Classification and statistics of finite index subgroups in free products[END_REF], Müller and Schlage-Puchta determine the statistics of given isomorphism types in free products and in [START_REF] Lubotzky | Subgroup growth and congruence subgroups[END_REF][START_REF] Goldfeld | Counting congruence subgroups[END_REF] Lubotzky-Goldfeld-Pyber and Lubotzky-Nikolov count the number of congruence subgroups in arithmetic groups. For an introduction to the subject of subgroup growth, we refer to the monograph by Lubotzky and Segal [START_REF] Lubotzky | Subgroup growth[END_REF].

In this paper we are interested in the growth of the sequence {e n (G)} n∈N . It is apparent that its behavior can be strikingly different from {s n (G)} n∈N . For example, we have just seen above that the sequence {s n (F 2 )} n∈N grows super exponentially whereas it is clear that the sequence {e n (F 2 )} n∈N is constantly equal to one. Similarly it is evident that for any surface group G the sequence {e n (G)} n∈N is constantly equal to one. These and free abelian groups are the only examples we know of (infinite, residually finite) groups with e n ≡ 1. To give an example where e n exhibits nontrivial growth let us describe its behaviour for Fuchsian groups (namely, discrete subgroup of PSL 2 (R)) containing torsion elements: we observe in Proposition 2.2 below that for a Fuchsian group G ⊂ PSL 2 (R) of finite covolume, e n (G) grows polynomially. For example, we obtain (using Müller and Schlage-Puchta's asymptotic mentioned above) that e n (PSL 2 (Z)) ∼ n 2 6 as n → ∞.

We note that this asymptotic contains geometric information about the orbifold PSL 2 (Z)\H 2 : in general, the exponent (2 in this case) equals the number of nontrivial divisors of the orders of its cone points. Moreover, in the case of PSL 2 (Z), the leading coefficient (1/6) appears as the inverse of the product these divisors.

To give our topic a more topological flavour, given a compact, connected manifold M let us denote by e n (M ) the number of homeomorphism types of connected degree d covering spaces of M . All the examples above are essentially 1-or 2-dimensional. For a topologist, after the case of Date: May 8, 2020. 2010 Mathematics Subject Classification. 57M10.

surfaces has been settled it is natural to next consider the case of 3-manifolds. A more pragmatic reason for this is that if M is a closed, orientable, aspherical 3-manifold, then we have e n (M ) = e n (π 1 (M )). This follows from the geometrization theorem, the rigidity theorem [START_REF] Mostow | Quasi-conformal mappings in n-space and the rigidity of hyperbolic space forms[END_REF], and work of Waldhausen [START_REF] Waldhausen | On irreducible 3-manifolds which are sufficiently large[END_REF] and Scott [START_REF] Scott | There are no fake Seifert fibre spaces with infinite π1[END_REF] (see e.g. [AFW15, Theorem 2.1.2]). Thus in this case the group-theoretical and topological problems are equivalent. Another is that the current understanding of 3-manifold groups (as opposed to higher-dimensional aspherical manifolds) is good enough to let us actually prove some nontrivial results.

For technical reasons it is often more convenient to study the following closely related sequence e n (M ) := sup{e k (M ) | k ≤ n}.

In Section 4.2 we point out that it follows from [START_REF] Agol | The virtual Haken conjecture[END_REF], [START_REF] Cooper | Essential closed surfaces in bounded 3-manifolds[END_REF], [START_REF] Burger | Counting hyperbolic manifolds[END_REF] and [BGLS10, Section 5.2] that e n (M ) grows as a power of a factorial for hyperbolic 3-manifolds. On the other hand it is clear that e n (M ) is a constant sequence for some choices of M , e.g. for the 3-torus and for 3-manifolds with cyclic fundamental group. As hyperbolic manifolds are in many ways generic among 3-manifolds this leads us to the following definition.

Definition 1.1. A manifold M is called exceptional if e n (M ) = 1 for all n ∈ N. In other words, M is exceptional, if for all n ∈ N any two connected degree n covers of M are homeomorphic.

Note that there is an analogous notion of exceptional groups, that is, a group is said to be exceptional if any two index n subgroups are isomorphic.

As a warm-up, in Lemma 2.1 we determine which compact 2-dimensional manifolds are exceptional (this makes for a surprisingly amusing exercise). In our main theorem we determine all exceptional compact 3-manifolds with empty or toroidal boundary.

Main Theorem. Let M be a compact 3-manifold with empty or toroidal boundary. Then M is exceptional if and only if it is homeomorphic to one of the following manifolds:

(1) k • S 1 × S 2 for k ≥ 1, (2) S 1 ×S 2 , (3) S 1 × D 2 , (4) T 2 × I, (5) T 3 , (6) all spherical manifolds except those with fundamental group P 48 × Z/p with gcd(p, 3) = 1 and p odd, or Q 8n × Z/q with gcd(q, n) = 1, q odd, and n ≥ 2.

In the theorem above and henceforth, S n denotes the n-sphere, D n denotes the n-dimensional disk, T n denotes the n-torus, I denotes the unit interval [0, 1], and k • M denotes the k-fold connected sum of the manifold M . The groups mentioned in item (6) are defined in Section 6. Some of the techniques we use in the proof of the main theorem apply in a a larger setting. For instance, as noted above, it follows from work of Agol et al that hyperbolic 3-manifolds are not exceptional. We also give a different proof of this fact that generalizes to lattices in most semisimple Lie groups as follows. (Recall that two Lie groups are called locally isomorphic if they have isomorphic Lie algebras.) Proposition 4.2. Let Γ be an irreducible lattice in a semisimple linear Lie group not locally isomorphic to PGL 2 (R). Then Γ is not exceptional.

Regarding PGL 2 (R) we noted (see above and Proposition 2.2) that Fuchsian groups with torsion exhibit nontrivial growth for e n and in particular are not exceptional. The latter is also true of Euclidean 2-orbifolds, and to prove this together with the fact that T 3 is the only exceptional Euclidean 3-manifold we use arguments which lead to the following generalization.

Proposition 5.2. Let E be a Euclidean space and Γ a lattice in Isom(E). Then Γ is exceptional if and only if it is free abelian.

Moreover for hyperbolic 3-manifolds we can produce regular non-homeomorphic covering spaces of equal degree: Proposition 4.5. Let Γ be the fundamental group of a complete hyperbolic 3-manifold of finite volume. Then there exist sequences c n , d n → +∞ such that for each n we can find at least c n normal subgroups of index d n in Γ, which are pairwise non-isomorphic.

1.1. Questions. There are many more questions about e n (M ) that arise naturally. 1.1.1. Quantitative questions. The only question we deal with in general is whether or not e n (M ) = 1 for all n ∈ N. Once it is known that this is not the case, it would of course be interesting to know how e n (M ) and e n (M ) behave as functions of n.

For any hyperbolic 3-manifold M , as we already noted above, there exists a constant α > 0 so that e n (M ) ≥ (n!) α , for all large enough n ∈ N. However, as of yet, there is no control on α in terms of, for instance, the hyperbolic geometry of M . We note that an upper bound on the smallest index of a subgroup of π 1 (M ) that surjects onto F 2 would give a lower bound on α.

On the other hand, our constructions of non-homeomorphic covers for Seifert fibered manifolds produce (in general) much sparser sequences of covers and we do not have an expectation for the answer to the quantitative question above.

The following is a sample of concrete questions we would like to know the answers to.

(1) Which growth types (polynomial, intermediate, exponential, factorial) appear for e n (M )?

(2) What are the 3-manifolds for which e n (M ) is polynomial, intermediate, exponential, factorial? (3) Are there hyperbolic 3-manifolds for which e n (M ) is not monotone even for large n? (4) Given a hyperbolic 3-manifold do e n (M ) and e n (M ) have the same growth type? (5) If M is a hyperbolic manifold, what information on the geometry of M is encoded in the sequence {e n (M )} n∈N ?

We note that for general 3-manifolds, many of the question above are also still open for the sequence {s n (π 1 (M ))} n∈N . For 2-dimensional manifolds we know more. For instance, it is known that for a hyperbolic 2-orbifold M , the sequence {s n (π 1 (M ))} n∈N does encode some geometric information: the factorial growth rate of s n (π 1 (M )) is linear in the area of the orbifold [START_REF] Müller | Character theory of symmetric groups and subgroup growth of surface groups[END_REF][START_REF] Liebeck | Fuchsian groups, coverings of Riemann surfaces, subgroup growth, random quotients and random walks[END_REF]. On the other hand, for hyperbolic 3-manifolds such a simple relation could never hold: there are hyperbolic 3-manifolds M of arbitrarily large volume but with rank(π 1 (M )) ≤ 5. The latter implies that the factorial growth rate of the sequence {s n (π 1 (M ))} n∈N cannot be more than 4. 1.1.2. Stronger versions of non-exceptionality. In most cases where we establish that a group G is not exceptional, we do so by providing, for infinitely many distinct values of d, pairs (G 1 , G 2 ) of non-isomorphic subgroups of G with index d in G. We are not aware of an example of an infinite residually finite group which is not exceptional, but for which this stronger property is in default.

1.1.3. Non-exceptionality for other classes of groups. One may inquire about the exceptionality, or lack thereof, of other interesting classes of groups. Here are some examples:

(1) The only exceptional right-angled Artin groups are the free groups and the free abelian groups1 . What about more general Artin groups?

(2) Which Coxeter groups are exceptional? It seems reasonable to expect that only a few finite ones are. (3) Are non-abelian polycyclic groups always non-exceptional? (4) Are there other examples of non-elementary word-hyperbolic groups, besides surface groups and free groups, that are exceptional?

Note that all the groups in the first three items are linear and finitely generated and hence, by Malcev's theorem [START_REF] Malcev | On isomorphic matrix representations of infinite groups[END_REF], residually finite. As such, they at least have infinitely many different finite index subgroups (given that they are infinite themselves).

1.2. Structure of the proof. Our proof consists of a case by case analysis. That is, we use the prime decomposition theorem and Perelman's work to divide up 3-manifolds into various geometric classes that we then address separately.

For most of the paper, we consider prime, orientable 3-manifolds. Other than certain simple cases (Proposition 3.2), we show that prime, compact, orientable 3-manifolds with non-empty toroidal boundary are not exceptional in Proposition 9.1. Among the closed, prime, orientable 3-manifolds, we see that S 1 × S 2 is clearly exceptional (Proposition 3.2), so it only remains to consider closed, irreducible, orientable 3-manifolds. Those with a trivial JSJ decomposition are divided into the following classes:

• hyperbolic 3-manifolds (Proposition 4.1),

• Euclidean 3-manifolds (Proposition 5.1),

• spherical 3-manifolds (Proposition 6.2),

• and the remaining Seifert fibered 3-manifolds (Proposition 7.4).

Then we treat closed, irreducible, orientable 3-manifolds with a non-trivial JSJ decomposition, where we need two separate arguments -one for Sol manifolds (Proposition 8.1) and one for all others (Proposition 9.1). The classification of prime non-orientable exceptional 3manifolds and non-prime exceptional 3-manifolds is an almost direct consequence of our work with prime and orientable 3-manifolds and is given in Section 10. The diagram in Figure 1 shows the structure of the proof. 1.3. Conventions. All manifolds are assumed to be compact and connected. We will call a compact manifold with non-empty toroidal boundary hyperbolic if its interior admits a complete hyperbolic metric of finite volume. Usually we do not distinguish between a manifold and its homeomorphism type. 1.4. Acknowledgements. SF is grateful for the support provided by the SFB 1085 "higher invariants" funded by the DFG. JR was supported by ANR grant ANR-16-CE40-0022-01-AGIRA. Most of the work on the paper was done while various subsets of the authors met at ICMAT Madrid, the Max Planck Institute for Mathematics in Bonn, the University of Bonn, and the University of Regensburg. We are very grateful for the hospitality of these institutions. Lastly, we thank the anonymous referee for the careful reading of the paper and for the useful comments and suggestions.

Surfaces and two-dimensional orbifolds

2.1. Surfaces. As a warm-up we state the following fairly elementary lemma, which doubles as a great exam problem in a first course on topology.

Lemma 2.1. The only exceptional compact 2-dimensional manifolds are the disk D 2 , the annulus, the Möbius band, the real projective plane RP 2 , and all closed orientable surfaces. Next assume that M is an orientable surface with at least one boundary component and that M is neither a disk nor an annulus. After possibly going to a finite cover we can assume that M has k ≥ 3 boundary components. By giving the boundary components the orientation coming from M , the boundary of M induces a summand of H 1 (M ), that is naturally isomorphic to

iff M ∼ = k • S 1 × S 2 , k > 1 (Proposition 10.3) Orientable Non-orientable Exceptional iff M ∼ = S 1 ×S 2 (Proposition 10.1) Irreducible M ∼ = S 1 × S 2 ⇒ exceptional (Proposition 3.2) Non-empty boundary M ∼ = T 2 × I or S 1 × D 2 ⇒ exceptional (Proposition 3.
k i=1 Z a i Z (a 1 + • • • + a k ).
Choose an epimorphism ϕ : π 1 (M ) → Z/k such that ϕ(a 1 ) = 1, ϕ(a 2 ) = -1, and ϕ(a i ) = 0 for i = 1, 2. On the other hand, we can also find an epimorphism ψ : π 1 (M ) → Z/k such ψ(a i ) = 0 for all i. But then the covers corresponding to ker(ϕ) and ker(ψ) have different numbers of boundary components, so they are not homeomorphic. Now suppose that M is a non-orientable surface that is not homeomorphic to either RP 2 or the Möbius band. There exists precisely one 2-fold cover of M that is orientable. On the other hand, we have H 1 (M ; Z/2) ∼ = (Z/2) k for some k ≥ 2. Since k ≥ 2, there exists at least one other 2-fold cover. This shows that M is not exceptional.

2.2. Fuchsian groups. If Γ is a finitely generated Fuchsian group of finite covolume, then the quotient O = Γ\H 2 is an orbifold of finite type, that is, a compact surface with genus g and k punctures as well as conical points with angles 2π/m i , 1 ≤ i ≤ s, for some g, s, k. We will call the tuple (g, k, m 1 , . . . , m s ) the signature of Γ. The orbifold Euler characteristic of O is defined by

-χ(O) := 2g -2 + k + s i=1 1 - 1 m i . It is multiplicative in covers, that is, if Γ is a finite index subgroup of Γ and O = Γ \H 2 then χ(O ) = [Γ : Γ ] • χ(O).
Proposition 2.2. Let Γ ⊂ PSL 2 (R) be a Fuchsian group of finite covolume with signature (g, k, m 1 , . . . , m s ). Let d be the number of distinct divisors of all m i . Then

(1)

n d-1 e n (Γ) n d-1 .
In specific cases one can say more by studying the proof of the upper and lower bounds more carefully; for example e n (PSL 2 (Z)) = n 2 /6 + O(n) as we stated in the introduction: to obtain this we note that e n (PSL 2 (Z)) is equal to the number of solutions to the equation (2) below by [START_REF] Müller | Classification and statistics of finite index subgroups in free products[END_REF]Theorem B]. Computing the covolume of the lattice appearing in the proof of the upper bound below (an exercise in Euclidean geometry) we get the result. We could have used Müller and Schlage-Puchta's result to get a more precise asymptote in more cases (though the computation of constants would require further analysis), but as it does not have the generality we require (in particular, it is not clear whether an analogous result holds for cocompact Fuchsian groups) we elected to give a more direct and elementary proof which works for all Fuchsian groups.

Proof. For the upper bound in (1) we note that two Fuchsian groups with respective signatures (g, k, m 1 , . . . , m s ) and (h, l, n 1 , . . . , n r ) are isomorphic if and only if (m 1 , . . . , m s ) = (n 1 , . . . , n r ) and either k = 0 = l and g = h or k, l > 0 and 2g + k = 2h + l. In addition, the condition that k = 0 is invariant under taking finite index subgroups.

Thus bounding the number of possible tuples (g + k, m 1 , . . . , m s ) for a degree n cover of O := Γ\H 2 with signature (g, k, m 1 , . . . , m s ) gives an upper bound for e n (Γ). Let Ω be the set of divisors of elements of the set {m i }. Let d denote the cardinality d := |Ω|. If Γ is a finite index subgroup of Γ with signature (g , k , m 1 , . . . , m t ) we further set r := 2g -2 + k , and

k m = |{1 ≤ i ≤ t : m i = m}|
for m ∈ Ω, so that (as each m i must divide some m i ) we have t i=1 1 -1

m i = m∈Ω, m =1 1 -1 m k m .
We see by multiplicativity of Euler characteristic that, letting n = |Γ/Γ | be the index, we have:

(2) r + m∈Ω m =1 1 - 1 m k m = nχ(O).
and hence the number of non-isomorphic subgroups of Γ is at most the number of points (r/n, k m /n : m ∈ Ω, m = 1) of the lattice 1 n Z d which belong to the (d -1)-simplex given by the intersection of the hyperplane x 0 + m∈Ω, m =1 (1 -1/m)x m = χ(O) with the positive quadrant. Now if P is a polygon in an affine space E and L a lattice in E we have |L ∩ nP | n dim E , and the upper bound follows 2 .

To prove the lower bound in (1) we construct explicit subgroups which contain distinct numbers of conjugacy classes of elements of order m i . To do this we use the following presentation for the Fuchsian group Γ of signature (g, k, m 1 , . . . , m s ):

(3) Γ = x 1 , . . . , x s , p 1 , . . . , p k , a 1 , b 1 , . . . , a g , b g x m i i , i x i j p j l [a l , b l ] .
Our first step is to reformulate our problem in terms of morphisms ρ : Γ → S n , where S n denotes the symmetric group on n letters. This will use the well known correspondence between subgroups of index n and transitive permutation representations ρ : Γ → S n (see for instance [START_REF] Lubotzky | Subgroup growth[END_REF]). We start with the following lemma. Given a group G, Z(x) will denote the centraliser of x ∈ G and x G will denote the conjugacy class of x in G.

Lemma 2.3. Let G be a group and ρ : G → S n a homomorphism with transitive image; let

H = Stab ρ {1} < G
be the stabiliser of 1 in G via ρ. Then the number of conjugacy classes in H containing an element of x G is equal to the cardinality of the quotient set ρ(Z(x))

{a ∈ {1, . . . , n}; ρ(x) • a = a} .

Proof. Write Z = Z(x). The orbits of the right-conjugation action of H on x G are in bijection with Z\G/H. The orbit map G/H → {1, . . . , n} is bijective (using transitivity) and Z-equivariant so Z\G/H ↔ ρ(Z)\{1, . . . , n}. Finally, the last map (bijectively) sends conjugacy classes of x lying in H to ρ(Z)-orbits fixed points of ρ(x).

We will distinguish between non-isomorphic finite index subgroups of the same index by comparing the numbers of conjugacy classes of maximal finite cyclic subgroups (on the orbifold side this corresponds to the cone points together with their orders). We will first deal with the cases where the algebraic structure of Γ is simpler, that is when we are in one of the following cases : k ≥ 2 (O has at least 2 cusps), k ≥ 1, g ≥ 1 (at least 1 cusp and genus at least 1) or g ≥ 2. We will deal with the remaining cases with a bootstrap argument.

Let m ∈ Ω, m = 1. Let ρ be a morphism Γ → S n and H = Stab ρ (1) as in the statement of the lemma above. Then if x ∈ H is a primitive element 3 of order m, we have that m is a divisor of some m i and x = gx

pm i /m i g -1 ∈ H for some p ∈ Z coprime to m, but gx m i /q i g -1 ∈ H for any m | q | m i , m < q.
If we fix such i the corresponding conjugacy classes of primitive order m elements correspond, via the lemma, to ρ(Z(x i ))-orbits on the set of m/m i -cycles of ρ(x i ) times the set of invertible elements of Z/mZ. When we consider only the maximal cyclic subgroups 2 More precisely |L ∩ nP | ∼ Cn dim P where C is the quotient of the volume of P by the covolume of L, so computing these volumes gives an explicit asymptotic upper bound.

3 Recall that a group element is primitive if it generates the maximal cyclic subgroups it lies in.

they generate we kill the (Z/mZ) × -factor; in conclusion, the number of conjugacy classes of maximal cyclic subgroups of H, which are of order m, is equal to the number of ρ(Z(x i ))-orbits on the set of m/m i -cycles of ρ(x i ).

Now in a Fuchsian group we have Z(x) = x for any nontrivial element. In particular, ρ(Z(x)) acts trivially on the cycles of ρ(x). So the number N of conjugacy classes of maximal cyclic subgroups of order m in H is given by the following formula :

(4) N = i: m|m i #{(m i /m)-cycles in the image of ρ(x i )}.
Let Ω i be the set of divisors of m i , and choose subsets 1

∈ Ω i ⊂ Ω i so that Ω \ {1} is the disjoint union of all Ω i \ {1} ; in particular we have (5) d -1 = s i=1 |Ω i | -1 Let n ≥ 2. Consider collections (π 1 , . . . , π s )
, where for all i, π i is a partition of n whose parts lie in {m i /m : m ∈ Ω i }. We claim (and will prove in the next paragraph) that for a positive proportion of such partitions it is possible to construct a representation of Γ in the symmetric group S n such that the cycle decomposition of ρ(x i ) corresponds to π i . Moreover, for different partition-tuples the subgroups cannot be conjugated to each other: if m ∈ Ω, m = 1 then maximal cyclic subgroups of order m can only come from lifts of the only x i such that m ∈ Ω i . It follows from (4) that the number of such subgroups is equal to the (m/m i )-part of π i , and these numbers determine the π i .

In this paragraph we prove the claim (under the hypotheses on g, k stated above). Assume first that k ≥ 2 or k ≥ 1 and g ≥ 1. Then we see from (3) that Γ decomposes as the free product

Γ = e 1 * • • • * e s * F
where F is a free group. Then any collection is realisable, as we can pick any image we want for each e i and then complete it by a transitive representation of F . In the remaining cases where k = 0 and g ≥ 2 then we can do the same thing, choosing ρ(a 1 ), ρ(b 1 ) to have transitive image and so that the product σ = ρ(e i )[ρ(a 1 ), ρ(b 1 )] is alternating, and finally a 2 , b 2 so that the relation in (3) holds (this is possible because every alternating permutation can be written as a commutator).

For a finite subset A ⊂ N denote by p A (n) the number of partitions of n into elements of A. The representations constructed above give rise to p Ω i (n) isomorphy classes of index n subgroups of Γ. Thus we have proven that for any n we have:

e n (Γ) ≥ s i=1 p Ω i (n).

It is an easily observed fact that p A (n)

n |A|-1 (see also [START_REF] Nathanson | Elementary methods in number theory[END_REF] for a more precise asymptotic), so that we get from the previous inequality that

e n (Γ) n s i=1 (|Ω i |-1)
and using (5) we can conclude that e n (Γ) n d-1 .

To finish the proof in the remaining cases (that is, those that do not satisfy our assumptions on k and g), we pass to a finite cover O which satisfies the hypotheses we used above and contains torsion elements of (at least) the same orders as O (see next lemma). Let f be the degree of this cover, then we get from what we previously proved that e f d (Γ) f d-1 for all n, which immediately implies that e n (Γ) n d-1 .

The following fact that we used at the end of the proof might be standard (and using results of Liebeck-Shalev our proof is almost immediate) but we did not find a reference for it.

Lemma 2.4. Let Γ be a Fuchsian group. Then it has a finite index subgroup H whose signature (g, k, . . .) satisfies either k ≥ 2 or g ≥ 2 and such that for any m if Γ contains an element of order m the same goes for H.

Proof. Recall that the group Γ admits a presentation of the form (3). In [START_REF] Liebeck | Fuchsian groups, coverings of Riemann surfaces, subgroup growth, random quotients and random walks[END_REF] Liebeck -Shalev prove the following: let C i be the conjugacy class of an element of S n which has only m i -cycles and f i fixed points, with f i bounded independently of n.

Let C = (C 1 , . . . , C s ) and Hom C (Γ, S n ) the set of homomorphisms ρ : Γ → S n such that ρ(x i ) ∈ C i (recall from (3) that x i is an element of order m i in Γ). Then |Hom C (Γ, S n )| ∼ (n!) 1-χ(Γ) • n O(1)
(loc. cit., Theorem 3.5) and an element in Hom C (Γ, S n ) is transitive with probability tending to 1 as n → +∞ (loc. cit., Theorem 4.4).

We choose all f i such that 1 ≤ f i ≤ m i (there is a unique possible choice for each n and i). Let ρ ∈ Hom C (Γ, S n ) be transitive (such ρ exists by the results presented above) and let H be the subgroup Stab ρ {1} and O the corresponding orbifold cover of O, with genus g and k cusps. Then we have that

2g + k = nχ(O) - s i=1 1 - 1 m i f i + 2 = nχ(O) -O(1)
and the right-hand side goes to infinity, so taking n large enough we can ensure 2g + k ≥ 4, which implies either k ≥ 2 or g ≥ 2. Moreover, since f i ≥ 1 if follows from Lemma 2.3 that the group H contains an element of order m i for each i. So this H satisfies the conclusion of the lemma.

Preliminaries

Let us start our discussion of 3-manifolds with some preliminary observations. Recall that a group G is called residually finite if

H G [G:H]<∞ H = {e},
where e ∈ G denotes the unit element. It follows from work of Hempel [START_REF] Hempel | Residual finiteness for 3-manifolds[END_REF], together with the proof of the geometrization theorem, that the fundamental group of a 3-manifold has this property: Theorem 3.1. [START_REF] Hempel | Residual finiteness for 3-manifolds[END_REF] Let M be a compact 3-manifold, then π 1 (M ) is residually finite.

Next, we make some elementary observations about certain simple 3-manifolds.

Proposition 3.2. The 3-manifolds S 3 , T 3 , T 2 ×I, S 1 ×D 2 , S 1 ×S 2 , and S 1 ×S 2 are exceptional. The twisted I-bundle over the Klein bottle is not exceptional.

Proof. It is an elementary exercise to verify that the manifolds mentioned in the first sentence are exceptional. For example, note that any manifold with cyclic fundamental group is exceptional. Finally note that the twisted I-bundle over the Klein bottle has two 2-fold covers, one of which is again homeomorphic to the twisted I-bundle over the Klein bottle and the other is homeomorphic to T 2 × I. Thus it is not exceptional.

We conclude the section with the following elementary observation, which uses the fact that our covers need not be regular.

Lemma 3.3. If a manifold M has a finite-sheeted cover p : M → M such that M is not exceptional, then M is not exceptional.

The hyperbolic case

In this section, we prove the following result.

Proposition 4.1. Hyperbolic 3-manifolds of finite volume are not exceptional.

As mentioned in the introduction, this follows by combining largeness ([Ago13]) with either [START_REF] Zimmermann | A note on hyperbolic 3-manifolds of the same volume[END_REF] or [START_REF] Burger | Counting hyperbolic manifolds[END_REF][START_REF] Belolipetsky | Counting arithmetic lattices and surfaces[END_REF], which yields a much stronger quantitative result. We also provide an independent proof which does not use largeness, and works in the more general setting of irreducible lattices in almost all semisimple Lie groups (Proposition 4.2). Finally we show that in any hyperbolic 3-manifold group one can also find non-isomorphic normal subgroups with the same index (Proposition 4.5).

4.1. Non-exceptionality of lattices in Lie groups. Let G be a semisimple Lie group and let X be the symmetric space associated to G (for example, G = PGL 2 (C) and X = H 3 ). Then for any discrete subgroup Γ ≤ G, the quotient Γ\X is a complete Riemannian orbifold locally isometric to quotients of X by finite subgroups of G (in particular, if Γ is torsion-free then Γ\X is a manifold). We will call such orbifolds X-orbifolds.

The Mostow-Prasad rigidity theorem [START_REF] Mostow | Quasi-conformal mappings in n-space and the rigidity of hyperbolic space forms[END_REF][START_REF] Mostow | Strong rigidity of locally symmetric spaces[END_REF][START_REF] Prasad | Strong rigidity of Q-rank 1 lattices[END_REF][START_REF] Margulis | Discrete subgroups of semisimple Lie groups[END_REF] states that if G is not locally isomorphic to PGL 2 (R), then two irreducible lattices Γ 1 and Γ 2 in G are isomorphic as abstract groups if and only if the orbifolds Γ i \X are isometric to each other. In particular the metric invariants of Γ\X are an isomorphism invariant of Γ. We will be using the systole to distinguish between subgroups: given an X-orbifold M this is defined as the infimum of lengths of closed geodesics on M , and we will denote it by sys(M ). Note that it follows from the Margulis lemma that sys(M ) is positive if M has finite volume.

The systole of Γ\X can be computed from the action of Γ on X. If g ∈ G is an element whose semisimple part does not belong to a compact subgroup of G, then the minimal translation Note that if Γ is cocompact, then Γ ah is the set of semisimple elements of infinite order in Γ.

We will now prove the following result, of which Proposition 4.1 is a special case.

Proposition 4.2. Let Γ be an irreducible lattice in a semisimple linear Lie group not locally isomorphic to PGL 2 (R). Then Γ is not exceptional.

Proof. Let G be a semisimple linear Lie group as in the statement, so we may assume that G < GL d (R) for some d and let Γ be a lattice in G. It is a standard consequence of local rigidity of Γ, which holds under the condition that G not be locally isomorphic to PGL 2 (R), that we may conjugate G so that there exists a number field F such that Γ < GL d (F ) (the proof given for [MR03, Theorem 3.1.2] in the cocompact case adapts immediately to all other groups). Let H be the Zariski closure of Γ in the Q-algebraic group obtained by Weil restriction of the linear F -algebraic group GL d (F ) to Q. By passing to a finite index subgroup if necessary, we may assume that every finite index subgroup of Γ has Zariski closure equal to H. Indeed, every chain of finite index algebraic subgroups . . . < Γ i+1 < Γ i < . . . < Γ is necessarily finite. So, a chain of finite index subgroups so that the Zariski closures are strictly contained in each other necessarily terminates after a finite number of steps and we may take the last term.

By finite generation of Γ there exists a finite set S of rational primes such that Γ ⊂ H Z[p -1 , p ∈ S] . For the rest of the proof we will fix a rational prime q ∈ S. Thus we can define the group of Z q -points, H q = H(Z q ). Nori-Weisfeiler strong approximation [START_REF] Weisfeiler | Strong approximation for Zariski-dense subgroups of semisimple algebraic groups[END_REF] implies that we can choose q so that the closure of Γ in H q is of finite index. Since H q is q-adic analytic we may assume that it is a uniform pro-q subgroup (cf. [DdSMS99, Theorem 8.1]), replacing Γ by a finite index subgroup if necessary. Now we prove the following lemma.

Lemma 4.3. Let p be a prime, H a uniform pro-p group, and γ ∈ H. There exists a sequence

(H 1 (k), H 2 (k)) of pairs of open subgroups of H such that |H/H 1 (k)| = |H/H 2 (k)|, H i (k + 1) ⊂ H i (k) and k≥1 H 1 (k) = {e}, k≥1 H 2 (k) = γ .
Proof. Let P k (H) be the lower p-series of H (see [DdSMS99, Definition 1.15]). Replacing H by some P k (H) we may assume that γ ∈ H \ P 2 (H). Uniformity of H implies that, independently of k ≥ 1, the group P k (H)/P k+1 (H) is an F p -vector space of fixed dimension c so that

|H/P k+1 (H)| = p ck .
On the other hand, we have γ p k ∈ P k+1 (H) \ P k+2 (H), so we get that

|H/ ( γ P k+1 (H))| = p (c-1)k .
We define

H 2 (k) = γ P c•k+1 which satisfies that H 2 (k + 1) ⊂ H 2 (k) and k≥1 H 2 (k) = γ . Let also H 1 (k) = P (c-1)•k+1
Then we have that

|H/H 1 (k)| = p (c-1)k = p ck /p k = |H/H 2 (k)|.
On the other hand, P k-l k +1 (H) ⊃ H 1 (k) so that k≥1 H 1 (k) = {e}. Now let γ ∈ Γ be a semisimple element in Γ of infinite order. Applying the lemma to H q and γ we get two sequences of subgroups

Γ 1 (q, k) = Γ ∩ H 1 (k + 1), Γ 2 (q, k) = Γ ∩ H 2 (k + 1)
which satisfy the same properties as H i . In particular, for any finite set Σ ⊂ Γ \ {1}, we have Σ ∩ Γ 1 (q, k) = ∅ for large enough k. Applying this to the finite sets

Σ R = {γ ∈ Γ ah : (γ) ≤ R}
with R going to infinity we see, using the formula (6), that:

lim k→+∞ sys(Γ 1 (q, k)\X) = +∞.
On the other hand, we have γ ∈ Γ 2 (k) for all k > 0 and it follows that ∀k > 0 : sys(Γ 2 (q, k)\X) ≤ (γ) and in particular, for any large enough k, the systoles of the X-orbifolds Γ 1 (q, k)\X and Γ 2 (q, k)\X are different. It finally follows from Mostow's rigidity theorem, as observed before the proposition, that the subgroups Γ 1 (q, k) and Γ 2 (q, k), which have the same index, cannot be isomorphic to each other for large enough k.

4.2.

Agol-Wise's theorem and a quantitative result. Hyperbolic 3-manifolds have finite degree covers with positive Betti number, which was proved by Agol, based on the work of Kahn-Markovic, Wise, and many others. In fact more is true; we have the following properties (see [START_REF] Agol | The virtual Haken conjecture[END_REF]Theorem 9.2] for the closed case and [CLR97, Theorem 1.3] for the case with toroidal boundary), in order of strength.

Theorem 4.4. Let M be a hyperbolic 3-manifold with finite volume. Then there exists:

(1) a finite cover M → M so that b 1 M > 0;

(2) for any r ≥ 1, a finite cover M → M so that b 1 M ≥ r;

(3) a finite cover M → M so that π 1 (M ) surjects onto a non-abelian free group.

We note that (1) can be used to give a proof of Proposition 4.1 which is similar but simpler than that of Proposition 4.2: if H 1 (Γ) contains a class φ of infinite order then the systole of the index n subgroup Γ n := φ -1 (nZ) stays bounded as n → +∞. As Γ is residually finite, there exists a sequence of subgroups Γ m whose systoles tend to infinity. For m large enough it thus follows from Mostow rigidity that the subgroups Γ An important step in the proof is the following special case.

Proposition 4.6. Let Γ be a lattice in a simple Lie group G, not isogenous to PGL 2 (R). Assume that b 1 (Γ) ≥ 2 and, for all n ≥ 1, let c n be the maximal number of pairwise non-isomorphic normal subgroups Γ Γ with Γ/Γ ∼ = Z/n. Then

lim inf n→+∞ c n n > 0.
We note that the hypothesis on Γ implies that G is of real rank 1, and in fact isogenous to one of SO(n, 1) or SU(n, 1), as lattices in higher-rank simple Lie groups have property (T) and hence finite abelianisation, as do those in Sp(n, 1) and the exceptional rank 1 group F -20 4 . 4.3.1. Remarks.

• Proposition 4.6 shows that when b 1 (Γ) ≥ 2, for any large enough n there exists a pair of non-isomorphic normal subgroups of index n within Γ. The conclusion of Proposition 4.5 is much weaker, and we do not know whether in general there are non-homeomorphic normal covers for every degree in a subset of N of natural density one. Note that in general this cannot be true of every degree-for example if M is a homology sphere then it cannot have regular covers of any prime degree.

• We still have some control over the density of the sequence d n in Proposition 4.5: it follows from the proof that we have d n n M where M = r!, with r the smallest index of a normal subgroup with b 1 ≥ 2.

• Moreover, the proof of Proposition 4.5 shows that we can take c n d

1/e-ε n for all ε > 0.

• The only ingredient specific to dimension 3 in the proof of Proposition 4.5 is property (2).

We note that this property holds for many lattices in higher dimensions as well (in particular all known lattices in SO(n, 1) in even dimensions), and for some complex hyperbolic lattices (see for example [START_REF] Marshall | Endoscopy and cohomology growth on U (3)[END_REF]).

• In [START_REF] Zimmermann | A note on hyperbolic 3-manifolds of the same volume[END_REF], Zimmermann produces a similar set of subgroups in Γ. In that construction, the quotients are isomorphic to Z/p n-i Z⊕Z/p i Z where p is some large prime. Moreover, the number of subgroups in Zimmermann's construction is sublinear as a function of their index.

4.3.2. Proof of Proposition 4.6. Let ϕ 2 (n) be the number of surjective homomorphisms from (Z/n) 2 to Z/n and let ϕ(n) be Euler's totient. Observe that (1, 0) → a, (0, 1) → * defines ϕ(n)×n surjective homomorphisms from (Z/n) 2 to Z/n, where (a, n) = 1 and * is any arbitrary element of Z/n. Similarly, there are n × ϕ(n) surjective homomorphisms defined by (1, 0) → * , (0, 1) → b, where (b, n) = 1 and * is an arbitrary element of Z/n. Counted without repetition, we have produced

ϕ(n) × n + n × ϕ(n) -ϕ(n) × ϕ(n) = (2n -ϕ(n))ϕ(n) surjective homomorphisms from (Z/n) 2 to Z/n. Thus, ϕ 2 (n) ≥ (2n -ϕ(n))ϕ(n).
Let h n be the number of surjective morphisms from Γ to Z/n. Since Γ surjects onto (Z/n) 2 we have that h n ≥ ϕ 2 (n). Since two surjective morphisms π 1 , π 2 : Γ → Q have the same kernel if and only if there exists an automorphism ψ of Q such that π 2 = ψ • π 1 , and the number of automorphisms of Z/n equals ϕ(n) and hence

h n |Aut(Z/n)| ≥ ϕ 2 (n) ϕ(n) ≥ 2n -ϕ(n) ≥ n
we get that there are pairwise distinct normal subgroups A 1 , . . . , A n ≤ Γ such that Γ/A j ∼ = Z/n for 1 ≤ j ≤ n.

By Mostow rigidity, we have that c n is at least the maximal number of A j which are pairwise non-conjugate in G. We want to prove that there exists b > 0 depending only on Γ such that for every n at most b among the A j s can be conjugated to each other, which implies that c n ≥ n/b and finally the conclusion of Proposition 4.6. For this we use a refinement of the arguments of [START_REF] Burger | Counting hyperbolic manifolds[END_REF] and [START_REF] Belolipetsky | Counting arithmetic lattices and surfaces[END_REF] that we mentioned in the previous subsection.

First we deal with the case where Γ is non-arithmetic: then an immediate and well-known consequence of Margulis' commensurator criterion for arithmeticity4 is that there is a unique maximal lattice Ω ⊂ G in the commensurability class of Γ, which is equal to the commensurator of Γ. Thus any g ∈ G which conjugates two A j s must belong to Ω, and since the A j s are normal in Γ each has at most b = |Ω/Γ| conjugates among them. Now assume Γ is arithmetic. By definition of arithmeticity there exists a semisimple algebraic group G defined over Z such that Γ ⊂ G(Z) with finite index. For p a rational prime let Z p denote the p-adic integers. Then a congruence subgroup of Γ is a subgroup of the form Γ ∩ U where U is a finite index (equivalently, open) subgroup in p G(Z p ). If Λ is a finite index subgroup in Γ we denote by Λ cong the congruence closure of a subgroup Λ ⊂ Γ: this is the smallest congruence subgroup of G(Q) containing Λ; explicitely the congruence closure of Λ is equal to Γ ∩ V where V is the closure in p G(Z p ) of Λ.

Lemma 4.7. Let Γ be an arithmetic group. There exists finitely many congruence subgroups Γ 1 , . . . , Γ m with the following property: if Λ is a finite index normal subgroup in Γ such that Γ/Λ is abelian then Λ cong is equal to one of the Γ i .

Proof. If Γ/Λ is abelian then so is Γ/Λ cong so that it suffices to show that there are only finitely many congruence subgroups ∆ ≤ Γ such that Γ/∆ is abelian. Let Γ be the derived subgroup of Γ, which is a Zariski-dense in G (since G does not have abelian quotients). By Nori-Weisfeiler strong approximation [START_REF] Nori | On subgroups of GLn(Fp)[END_REF][START_REF] Weisfeiler | Strong approximation for Zariski-dense subgroups of semisimple algebraic groups[END_REF] it follows that the closure of Γ in p G(Z p ) has finite index in that of Γ. This means that at most finitely many congruence subgroups of Γ contain Γ , which is the statement we wanted to prove.

The commensurator of Γ is equal to (the image in G of) G(Q) and we have that gΛg -1 cong = gΛ cong g -1 for all g ∈ G(Q). It follows that if two subgroups of Γ are conjugate to each other an element conjugating them must belong to G(Q) and conjugate their congruence closures to each other as well: in particular, if the latter are equal then the element must belong to its normaliser. Let Γ 1 , . . . , Γ m be given by the lemma and let i be an index such that Γ i contains n ≥ n/m of the A j s, and assume for notational ease that those are A 1 , . . . , A k . It follows from the above that for any n, any element conjugating two of the A 1 , . . . , A k must belong to the normaliser Ω i of Γ i . Thus, as the A j are normal in Γ, the maximal number of conjugates among

A 1 , . . . , A k is c = |Ω i /(Γ i ∩ Γ)|.
In conclusion, we have shown that we can find at least k/C = n/(cm) among the A j that are pairwise not conjugate in PGL 2 (C), hence our claim follows (with b = cm).

Proof of Proposition 4.5. Let ∆ be a lattice in PGL 2 (C). By Theorem 4.4(2) there exists a finite index normal subgroup Γ ∆ with b 1 (Γ) ≥ 2, so that we may apply Proposition 4.6 to Γ. Let a 1 , . . . , a r be representatives for the left cosets of Γ in ∆. Let n ≥ 1 and B 1 , . . . , B cn the subgroups obtained in Proposition 4.6. Then since B j Γ we get, for all 1 ≤ j ≤ c n , that:

C j = r i=1 a i B j a -1 i is normal in ∆.
We recall that if A is a permutation group of degree r (that is, a subgroup of the symmetric group S r ) and B any group, the wreath product A B is the semidirect product A B r where A acts on B r by permuting indices.

Lemma 4.8. There exist r, l ∈ N, depending only on Γ and ∆, such that for each 1 ≤ j ≤ c n , there exists a finite abelian group Q j so that

(Z/n) l Q j and ∆/C j → S r Q j .
Proof. Let ρ be the morphism ∆ → S(∆/C j ) associated to the left-translation action. It respects the decomposition into left Γ-cosets and hence it has image inside S(∆/Γ) S(Γ/C j ). Moreover the stabiliser of a block has its image in a conjugate of the image of the action of Γ on Γ/C j .

It remains to see that Γ/C j is abelian of exponent n (we can then take l to be the rank of H 1 (Γ) ⊗ Z/n). To do so we need only remark that the subgroup [Γ, Γ] • Γ n < Γ generated by commutators and nth powers is characteristic in Γ and contained in B j , hence it is also contained in

C j . Since [Γ, Γ] • Γ n = ker (Γ → H 1 (Γ) ⊗ Z/n) this implies that H 1 (Γ) ⊗ Z/n Γ/C j
and hence that the image of ∆/C j in the second factor in the wreath product is a quotient of H 1 (Γ) ⊗ Z/n.

By the same argument as in the proof of the previous proposition we can eliminate some of the C j s so that at least b n = c n /a (where a depends only on Γ) are pairwise non-isomorphic. Indeed, if ∆ is non-arithmetic then the same argument applies verbatim, while if ∆ is arithmetic we have to show that for C j ∆ with ∆/C j → S r Q j there are only finitely many possibilities for the congruence closures of C j in ∆. To do this, we only need to note that this is true of the congruence closures of the C j in ∆ 1 := ker (Γ → S r ) .

Indeed, since all the ∆ 1 /C j are abelian, this follows from Lemma 4.7. Moreover, since these closures contain those in ∆ it is also true of the latter.

So we may assume that C 1 , . . . , C bn are pairwise non-conjugate, with b n ≥ c n /a ≥ n/a for some a ≥ 1 independent of n. A priori the C j have different indices in ∆. But by Lemma 4.8 the orders |∆/G j | all divide r! • n lr . Let δ(N ) denote the number of divisors of a positive integer N , then using the classical estimate that ∀ε > 0 there exists a constant K ε so that

δ(N ) ≤ K ε • N ε for all N ∈ N (see e.
(r! • n rl ) 1/(rl) we see that b n /δ(r! • d r n ) -----→ n→+∞ +∞.
Hence by the pigeonhole principle we see that as n → +∞ an unbounded number of the C j have the same index in ∆. This finishes the proof of Proposition 4.5.

Euclidean manifolds

The general argument we will present for Seifert fibered manifolds (Proposition 7.4) will not hold for manifolds that are finitely covered by either the 3-torus or the 3-sphere. Seifert fibered manifolds that are finitely covered by T 3 are exactly the closed Euclidean manifolds (see eg. [START_REF] Scott | The geometries of 3-manifolds[END_REF]). In this section, we prove the following proposition.

Proposition 5.1. The only closed orientable exceptional Euclidean 3-manifold is the 3-torus T 3 .

In fact we will prove a result about a more general class of groups. A group Γ is called a crystallographic group if it acts properly discontinuously and cocompactly on a finite-dimensional vector space; we refer to [START_REF] Auslander | An account of the theory of crystallographic groups[END_REF] for an overview. It then has to preserve a Euclidean metric on this space, so another definition is to say that crystallographic groups are lattices in isometry groups of Euclidean spaces.

In particular torsion-free crystallographic groups are the fundamental groups of finite-volume flat Riemannian manifolds, also called Euclidean manifolds. These are classified up to dimension 4. In dimension 2 there are only the torus and Klein bottle. It has been known since the 1930s that there are only ten closed 3-manifolds which are covered by the 3-torus, among which four are non-orientable [START_REF] Nowacki | Die euklidischen, dreidimensionalen, geschlossenen und offenen Raumformen[END_REF][START_REF] Hantzsche | Dreidimensionale euklidische Raumformen[END_REF]. The ten closed Euclidean 3-manifolds can be explicitly constructed [START_REF] Conway | Describing the platycosms[END_REF], and as such concrete geometric arguments can be used to show that none of these but the 3-torus are exceptional. However it is perhaps simpler to use a more algebraic argument to prove the following more general result. Proposition 5.2. Let Γ be a crystallographic group. Then Γ is exceptional if and only if it is free abelian.

Proof. Let E be a Euclidean vector space such that Γ is a lattice in Isom(E) ∼ = O(E) E. Let π be the map from Γ to O(E). Then T = ker(π) is a free abelian group of rank dim(E) by Bieberbach's theorem. In the sequel we will identify it with a subgroup of E. From now on we will assume that Γ is not free abelian, so π has nontrivial image.

Let Π = π(Γ), let p be a prime dividing the order |Π| and let m = |Π|/p. Choose an element σ ∈ Π which has order p.

Lemma 5.3. Let E f = ker(σ -Id). Then T = (T ∩ E f ) ⊕ (T ∩ E ⊥ f )
is a finite index subgroup in T .

Proof. It suffices to show that T has the same rank as T , in other words, that T ∩ E f and

T ∩ E ⊥ f are of respective rank dim(E f ) and dim(E ⊥ f ). Let ω = (Id + σ + • • • + σ p-1 )| T .
We recall that σT ⊂ T : if γ ∈ Γ is any element projecting to σ and v ∈ T , a quick computation shows that γvγ -1 = σv and hence σv ∈ T . On the other hand, ω is equal to p times the orthogonal projection onto E f and we have that ωT ⊂ (T ∩ E f ). So the image is discrete and has rank at most dim(E f ), and the kernel has rank at most dim(E ⊥ f ) = dim(E) -dim(E f ). Thus, both inequalities must be equalities and this finishes the proof.

Let

A = π -1 σ . Since σ preserves E f , E ⊥
f , and T we have that σT = T and hence, using the semi-direct product structure, that the subgroup T is normal in A. Likewise, we see that the subgroup

T = (T ∩ E f ) ⊕ p(T ∩ E ⊥ f )
is also normal in A and as we have σ = Id we have dim(E ⊥ f ) = r > 0 and we get that

|T /T | = p r |T /T |. Now choose any subgroup L of index p r-1 |T /T | in T , γ ∈ Γ such that π(γ) = σ and let B = γ T so that B/T = Z/p (because γ p ∈ E f ∩ T ). It follows that |Γ/B| = m/p • |A/B| = m/p • p r |T /T | = mp r-1 |T /T | = m|T /L| = |Γ/L|
which finishes the proof: indeed, B is not abelian while L is and it follows that they cannot be isomorphic.

Proof of Proposition 5.1. Recall that 3-manifolds with abelian fundamental group are wellunderstood; see e.g. [AFW15, Table 2]. Observe that if a closed Euclidean 3-manifold M has free abelian π 1 (M ), then π 1 (M ) ∼ = Z 3 and M ∼ = T 3 , which we know is exceptional from Proposition 3.2.

Spherical manifolds

Seifert fibered manifolds finitely covered by the 3-sphere, namely the spherical manifolds, also require a different proof than the general case. We will soon completely classify the exceptional spherical 3-manifolds (Proposition 6.2). However, we will first need some notation.

It is known that spherical 3-manifolds are exactly the quotients of S 3 by finite groups that act by isometries [START_REF] Scott | The geometries of 3-manifolds[END_REF]. These quotients of S 3 have been classified by Hopf [Hop26, Section 2] (see also [START_REF] Aschenbrenner | 3-manifold groups[END_REF]p. 12] and [Mil57, Theorem 2]) as follows (note that the group Q 4n (in the notation of [AFW15, p. 12]) is isomorphic to D 2n when n is odd).

Theorem 6.1. The fundamental group of a spherical 3-manifold is of exactly one of the following forms:

• The trivial group, • Q 8n := x, y| x 2 = (xy) 2 = y 2n , for n ≥ 1,
• the binary octahedral group: P 48 := x, y| x 2 = (xy) 3 = y 4 , x 4 = 1 , • the binary icosahedral group:

P 120 := x, y| x 2 = (xy) 3 = y 5 , x 4 = 1 , • D 2 m (2n+1) := x, y| x 2 m = 1, y 2n+1 = 1, xyx -1 = y -1 , for m ≥ 2 and n ≥ 1, • P 8•3 m := x, y, z| x 2 = (xy) 2 = y 2 , zxz -1 = y, zyz -1 = xy, z 3 m = 1 , for m ≥ 1,
• the direct product of any of the above groups with a cyclic group of relatively prime order.

The subscripts in the notation for the groups above always denote their order. We are now ready to state the main result of this section. Proposition 6.2. A spherical manifold is exceptional if and only if its fundamental group is of one of the following forms:

• The trivial group, • Q 8 , • P 120 , • D 2 m (2n+1) for m ≥ 2 and n ≥ 1, • P 8•3 m for m ≥ 1,
• the direct product of any of the above groups with a cyclic group of relatively prime order.

Before giving the proof, we gather a few relevant facts. Recall that the fundamental group determines a spherical manifold unless it is cyclic and non-trivial (see [START_REF] Orlik | Seifert manifolds[END_REF]p. 133]). This fact, for the larger class of closed, irreducible 3-manifolds, yields the following lemma (see [AFW15, Theorem 2.1.2]). Lemma 6.3. Let M be a closed, orientable, irreducible 3-manifold. Let G and H be finite index subgroups of π 1 (M ) and let M G and M H be covers of M that correspond to G and H, respectively. Suppose G is not a finite cyclic group, then G and H are isomorphic if and only if M G and M H are homeomorphic.

The case of spherical manifolds with cyclic fundamental groups, namely, lens spaces, is more subtle. We recall the following well known result of Reidemeister [START_REF] Reidemeister | Homotopieringe und Linsenräume[END_REF]. Theorem 6.4. Let L(p, q) and L(p, q ) be two lens spaces. Then L(p, q) and L(p, q ) are homeomorphic if and only if q ≡ ±q ±1 mod p.

We are now ready to prove Proposition 6.2.

Proof of Proposition 6.2. For each of the manifolds listed in Theorem 6.1, we will follow one of the following two strategies.

• In order to show that a given manifold is not exceptional, we will show that its fundamental group has two non-isomorphic subgroups with the same index. • In contrast, in order to show that a given manifold is exceptional, we will first show that its fundamental group has a unique isomorphism type of subgroup with any fixed index. By Lemma 6.3, this implies that it only remains to consider the case when the subgroups of a given index are all isomorphic to a fixed finite cyclic group. In this case, we will show that corresponding covers are homeomorphic, either by using Theorem 6.4 or by showing that these subgroups are conjugate to each other.

We divide our proof into multiple lemmata. In what follows we will often tacitly identify a manifold with its fundamental group.

First we note: Lemma 6.5. Any spherical 3-manifold with cyclic fundamental group (namely, a lens space) is exceptional.

Proof. This follows since a cyclic group contains at most one subgroup of a given index.

Lemma 6.6. The spherical 3-manifold with fundamental group P 48 is not exceptional while that with fundamental group P 120 is exceptional.

Proof. The proper subgroups of P 48 and P 120 are well known (see e.g. [START_REF] Gonçalves | The classification of the virtually cyclic subgroups of the sphere braid groups[END_REF]Appendix]). The group P 48 has Z/8 and Q 8 as proper subgroups, and hence, the spherical manifold with fundamental group P 48 is not exceptional. The group P 120 has at most one isomorphism type of subgroup of any given index, and those of order 2, 3, 4, 5, 6, and 10 are isomorphic to finite cyclic groups. Note that by Theorem 6.4 there is a unique 3-manifold with fundamental group with order 2, 3, 4, or 6. Any two proper subgroups of P 120 of order 5 are Sylow 5-subgroups and hence conjugate to each other. Also, it is known that any order two element of P 120 is contained in the center. Since the order 5 subgroups are conjugate, this implies that the order 10 subgroups are also conjugate to one another. Thus, we see that the spherical manifold with fundamental group P 120 (namely the Poincaré homology sphere) is exceptional.

All that remains are the three infinite sequences and products with cyclic groups of coprime order. We start with the groups Q 8n : Lemma 6.7. The spherical 3-manifold with fundamental group Q 8n is exceptional if and only if n = 1.

Proof. The group Q 8 is the quaternion group, the proper subgroups of which are all cyclic with either order 2 or 4. Using a similar argument as above, one can show that the spherical manifold with fundamental group Q 8 is exceptional. Next, we prove that for n > 1, the group Q 8n is not exceptional. In particular, we will show that these groups have two non-isomorphic subgroups of index 2. Let N 1 = y and N 2 = x be the subgroups normally generated by y and x respectively. It is easy to verify that both N 1 and N 2 are subgroups of index 2. Note that N 1 is cyclic since xyx -1 = y -1 in Q 8n . Since N 1 has index 2, the order of N 1 is 4n and in particular, y has order 4n in Q 8n . On the other hand, we will show that N 2 is non-abelian. Using the x = yxy relation, it is easy to see that yxy -1 • x -1 = y 2 ∈ N 2 . Now suppose that y 2 xy -2 = x. Since x 2 = (xy) 2 , and hence x = yxy, we see that y 4 x = x, and thus, that the order of y is at most 4, which contradicts our previous observation, since n > 1. Thus, the groups Q 8n for n > 1 are not exceptional.

For the dihedral groups we have: Lemma 6.8. The spherical 3-manifold with fundamental group D 2 m (2n+1) are exceptional for all m ≥ 2 and n ≥ 1.

Proof. We will invoke some Sylow theory. First, note that the subgroup generated by x ∈ D 2 m (2n+1) is a Sylow 2-subgroup of D 2 m (2n+1) and is isomorphic to Z/2 m . It is also easy to show that the abelianization of D 2 m (2n+1) is isomorphic to Z/2 m , and is generated by the image of x. Let N 1 = y be the subgroup of D 2 m (2n+1) normally generated by y. Note that N 1 is cyclic, since xyx -1 = y -1 . Thus, we have the following exact sequence of groups

1 → N 1 ∼ = Z/(2n + 1) → D 2 m (2n+1) → Z/2 m ∼ = x → 1
corresponding to the abelianization map. Further, let N 2 = y, x 2 . Since x 2 commutes with y, we have the following exact sequence

1 → N 2 ∼ = Z/(2n + 1) × Z/2 m-1 → D 2 m (2n+1) → Z/2 ∼ = {0, x} → 1.
Clearly, N 2 is cyclic of order 2 m (2n + 1) and is generated by yx 2 .

Let Γ ≤ D 2 m (2n+1) be a subgroup and φ be the restriction to Γ of the homomorphism D 2 m (2n+1) → Z/2 in the last sequence. We first consider the case where φ is a surjection. Then Γ must also surject onto Z/2 m in the abelianization map, since every map from D 2 m (2n+1) to an abelian group factors through the abelianization. There are two options -either |Γ| = 2 m or |Γ| > 2 m . In the first case, Γ is cyclic and conjugate to x by Sylow's theorem. If |Γ| > 2 m , we now show that Γ must be isomorphic to D 2 m (2n +1) for some n ≤ n.

First, note that due to the relation xyx -1 = y -1 , any element of D 2 m (2n+1) can be written as y i x j for some 0 ≤ i ≤ 2n and 0 2n+1) . Since (2n+1, 2 m-1 ) = 1, x 2(2n+1) generates x 2 . As a result, x 2 ∈ y d x 2d , and thus, y d ∈ y d x 2d . Next, let y i x j be an element of Γ such that y i x j / ∈ y d x 2d . If no such element exists, then Γ ≤ N 2 which is a contradiction. We show that j is odd, as follows. Suppose for the sake of contradiction that j is even. Since x 2 ∈ y d x 2d and y i x j / ∈ y d x 2d , y i / ∈ y d x 2d . Since y d ∈ y d x 2d , this implies that i is not a multiple of d. In particular, gcd(i, d) < d. Moreover, since y i ∈ Γ and y d ∈ Γ, y gcd(i,d) ∈ Γ. However, note that the order of y gcd(i,d) is 2n+1 gcd(i,d) since the order of y in

≤ j ≤ 2 m -1. Recall that N 2 = yx 2 . Then ker φ = Γ ∩ N 2 = Γ ∩ yx 2 and thus, ker φ = (yx 2 ) d = y d x 2d for some d ≥ 1. Since [Γ : y d x 2d ] = 2 and | y d x 2d | = 2 m-1 (2n+1) d , we have |Γ| = 2n+1 d • 2 m . Since |Γ| > 2 m , we see that d < 2n + 1. Since Γ surjects onto Z/2 m , |Γ| is divisible by 2 m . As a result, d divides 2n + 1. This implies that d is odd and 2n+1 d is an integer. Note that (y d x 2d ) 2n+1 d = x 2(
D 2 m •(2n+1) is 2n + 1. Then, 2n+1 gcd(i,d) divides |Γ| = 2n+1 d • 2 m , which implies that d divides 2 m • gcd(i, d).
Since d is odd, it must divide gcd(i, d) < d which is a contradiction. Thus, j must be odd; denote j by 2k + 1 for some k.

We will now complete the proof by showing that the subgroup generated by y d and y i x 2k+1 is equal to Γ and isomorphic to D 2 m (2n +1) where 2n + 1 = 2n+1 d . We have the following identities

(y i x 2k+1 ) 2 m = (y i x 2k+1 y i x 2k+1 ) • • • (y i x 2k+1 y i x 2k+1 ) = x 2k+1 • • • x 2k+1 = (x 2k+1 ) 2 m = 1, (y d ) 2n+1 d = y 2n+1 = 1, (y i x 2k+1 )y d (y i x 2k+1 ) -1 = y i x 2k+1 y d x -2k-1 y -i = y i xy d x -1 y -i = y i y -d dy -i = y -d ,
where we have used the facts that x 2 y d x -2 = y d , xyx -1 = y -1 , and yxy = x. It is easy to check that any element of y d , y i x 2k+1 can be uniquely expressed in the form (y

d ) i (x 2k+1 ) j , where 0 ≤ i ≤ 2n+1 d -1 and 0 ≤ j ≤ 2 m -1 and thus, | y d , y i x 2k+1 | = |Γ| = 2n+1 d • 2 m which completes the argument.
When φ is not surjective, it must be the zero map and thus, Γ is a subgroup of the cyclic subgroup N 2 ∼ = Z/((2n + 1)2 m-1 ) which implies that it is cyclic.

Returning to the group D 2 m (2n+1) , we have now shown that subgroups of a given fixed index are isomorphic. More precisely, subgroups are of the form D 2 m (2n +1) for n ≤ n, or cyclic groups with order either 2 m , or a factor of (2n + 1)2 m-1 . The latter arose as subgroups of a cyclic group and thus, occur exactly once. We saw earlier that any subgroup of order 2 m is a Sylow 2-subgroup, and thus such subgroups are conjugate to one another. This concludes that spherical manifolds with fundamental group D 2 m (2n+1) are exceptional for m ≥ 2 and n ≥ 1.

For the sequence P 8•3 m we have: Lemma 6.9. The spherical 3-manifold with fundamental group P 8•3 m is exceptional for all m ≥ 1.

Proof. First note that P ab 8•3 m ∼ = Z/3 m . Since Q 8 ∼ = x, y < P 8•3 m lies in the kernel of the abelianization map and |Q 8 | = 8, it must actually coincide with this kernel. Thus this copy of Q 8 is a normal subgroup and as such is the unique Sylow-2 subgroup of P 8•3 m . We obtain the following exact sequence 1

→ Q 8 → P 8•3 m → Z/3 m → 1 corresponding to the abelianization map. Now let Γ ≤ P 8•3 m . If the image of Γ in the abelian- ization map is trivial, it needs to lie in Q 8 , hence it is either Q 8 , Z/2, or Z/4.
For the case when the image of Γ is non-trivial, we first show that Γ is either a finite cyclic group or Q 8 × Z/3 j for some 0 < j ≤ m. Since the order of Γ divides the order of P 8•3 m , we see that |Γ| = 2 i • 3 j for some 0 ≤ i ≤ 3, 0 ≤ j ≤ m. Since Γ has non-trivial image in P ab 8•3 m , we see that j = 0. With these restrictions, we see from the list in Theorem 6.1 that the only possible groups are D 4•3 j , D 8•3 j , P 8•3 j , Q 8•3 j , Q 8 × Z/3 j , Z/3 j , Z/2 • 3 j , Z/4 • 3 j , and Z/8 • 3 j , where 0 < j ≤ m.

It is straightforward to see that the abelianization of D 2 m •(2n+1) is Z/2 m , and the abelianization of Q 8•3 j is Z/2 × Z/2. Neither of these can surject onto Z/3 j . Next, suppose that 8 divides the order of Γ. Then Γ has a unique Sylow-2 subgroup, namely Q 8 , where the uniqueness follows from normality. Since the unique Sylow-2 subgroup of Z/8 • 3 j is cyclic, we see that Γ Z/8 • 3 j . So, if 8 divides the order of Γ, we see that Γ is either isomorphic to P 8•3 j or Q 8 × Z/3 j . In these cases Γ has a Sylow-3 subgroup, denoted by Γ 3 . Recall that there is a Sylow-3 subgroup of P 8•3 m which is a copy of Z/3 m generated by z. Since any Sylow-3 subgroup of Γ must be contained in some Sylow-3 subgroup of P 8•3 m and Sylow-3 subgroups of a given group are conjugate, there is some g ∈ P 8•3 m such that g -1 z 3 m-j g generates Γ 3 ≤ Γ, and thus, z 3 m-j ∈ gΓg -1 generates gΓ 3 g -1 . Note that the subgroup gΓg -1 ≤ P 8•3 m has order divisible by 8, so as before, its Sylow-2 subgroup coincides with the Sylow-2 subgroup of P 8•3 m , namely the copy of Q 8 generated by x and y. Since z 3 commutes with < x, y >, we see that < x, y, z 3 m-j > ∼ = Q 8 × Z/3 j ≤ gΓg -1 for j = m and in fact, Γ ∼ = Q 8 × Z/3 j due to cardinality. We have thus reduced the possibilities for Γ to Z/3 j , Z/2 • 3 j , Z/4 • 3 j , and Q 8 × Z/3 j , where 0 < j ≤ m when the image of Γ is non-trivial.

Thus, the possible subgroups of P 8•3 m are of the form Z/2, Z/4, Q 8 , Z/3 j , Z/2•3 j , Z/4•3 j , Q 8 × Z/3 j , where 0 < j ≤ m. Again, no two distinct isomorphism types of subgroups have the same order, therefore we only need to consider the case of cyclic subgroups. By Theorem 6.4 there is a unique 3-manifold with fundamental group Z/2 or with fundamental group Z/4. For Z/3 j , we know that this group is contained some Sylow-3 subgroup of P 8•3 m , which is cyclic. Hence any two subgroups with order 3 j are conjugate to each other, and thus, correspond to isomorphic covering spaces. Suppose Γ ∼ = Z/2 • 3 j ∼ = Z/2 × Z/3 j . Note there is a Sylow-3 subgroup of Γ which is contained in some Sylow-3 subgroup of P 8•3 m which is cyclic and conjugate to z , as we saw earlier. Hence there is some g ∈ P 8•3 m such that g -1 z 3 m-j g is contained in Γ. Similarly, any Sylow-2 subgroup of Γ corresponds to a subgroup of order two within the copy of Q 8 in P 8•3 m . There is a unique such subgroup, generated by x 2 . Thus, since Γ is cyclic, we see that Γ = x 2 , g -1 z 3 m-j g . Note that any two such subgroups are conjugate to each other since x 2 is central. Lastly, suppose Γ ∼ = Z/4 • 3 j . Let Γ 2 be a Sylow-2 subgroup of Γ. Then it is contained in Q 8 , namely the Sylow-2 subgroup of P 8•3 m , and consequently, it is either x , y , or xy . Further, let Γ 3 be a Sylow-3 subgroup of Γ. Then again g -1 z 3 m-j g is contained in Γ for some g ∈ P 8•3 m . We have now seen that gΓg -1 is either x, z 3 m-j , y, z 3 m-j , or xy, z 3 m-j and any two such subgroups are conjugate to each other since z 3 m-j is central, and x, y and xy are conjugates. This completes the proof.

Finally, we need to consider direct products with cyclic groups: Lemma 6.10. Let G be a group from the statement of Theorem 6.1 and C a finite cyclic group so that gcd(|G| , |C|) = 1. Then G × C is exceptional if and only if G is exceptional.

Proof. Subgroups of direct products of finite groups with relatively prime order are direct products of subgroups in the factors. Hence, since the orders of the groups in the direct product need to be relatively prime and cyclic groups are exceptional, taking a direct product with a cyclic group preserves being exceptional or not.

We have now addressed each case in Theorem 6.1 and thus, our proof is completed.

The general Seifert fibered case

In this section, we prove the following result.

Proposition 7.1. Closed orientable Seifert fibered 3-manifolds, other than those finitely covered by T 3 , S 1 × S 2 or S 3 , are not exceptional.

Note that along with Propositions 5.1 and 6.2, this shows that closed Seifert fibered 3manifolds are not exceptional.

Our proof is based on the following proposition, which can for instance be found in [AFW15, p. 52 (C.10)]: Proposition 7.2. Let M be a closed Seifert fibered 3-manifold. There exists a finite cover M → M so that M is an S 1 -bundle over a closed orientable surface.

In order to find distinct finite covers of a Seifert fibered manifold, it thus suffices to find distinct finite covers of S 1 -bundles over closed orientable surfaces. Intuitively, the way we produce these is to take covers both in the S 1 -direction and the surface direction. To make this idea precise, we will use the Euler number e(π) of our circle bundles to distinguish covers (see [START_REF] Scott | The geometries of 3-manifolds[END_REF]p. 427,436] for a definition). In particular, we will use the following property of Euler numbers (see for instance [Sco83a, Lemma 3.5]).

Lemma 7.3. Let d ∈ N and S 1 → M π → Σ be an S 1 -bundle over a closed oriented surface Σ, such that M is orientable. Moreover, let M be a degree d finite cover of M , so that M is the total space of the following bundle We now prove the following proposition. This, together with the fact that the only orientable S 1 -bundle over S 2 is S 1 × S 2 , will complete the proof of Proposition 7.1.

S 1 → M π → Σ.
Proposition 7.4. Let Σ be a closed orientable surface that is not a sphere and let M be an S 1 -bundle over Σ. Then M is exceptional if and only if M is the trivial S 1 -bundle over the 2-torus.

Proof. Our first claim is that the map π 1 (S 1 ) → π 1 (M ) is injective. This follows from the long exact sequence in homotopy of the fibration

. . . → π 2 (Σ) → π 1 (S 1 ) → π 1 (M ) → π 1 (Σ) → . . . .
Our assumption on the genus of Σ implies that π 2 (Σ) = {e} and hence that the map π 1 (S 1 ) → π 1 (M ) is injective.

First we assume the bundle is non-trivial. Let t ∈ π 1 (S 1 ) denote a generator. Residual finiteness of 3-manifold groups (see Theorem 3.1) implies that we can find a finite group G and a surjection ϕ : π 1 (M ) → G so that

ϕ(t) = e.
Let us denote the induced degree d = |G| cover by M → M . Since t is mapped to a non-trivial element, the induced circle cover is non-trivial. Then Lemma 7.3 tells us that the induced

S 1 -bundle S 1 → M π → Σ satisfies |e( π)| < d • |e(π)| .
To build the second cover, take any degree d surface cover Σ → Σ (these exist for any d) and pull back the S 1 -bundle. This gives rise to a degree d cover M → M , that has the structure of a S 1 -bundle S 1 → M π → Σ. Applying Lemma 7.3 again, we obtain

|e( π)| = d • |e(π)| .
Since our bundle is non-trivial, we have e(π) = 0. It can now for instance be extracted from the Gysin sequence that if S 1 → N → Σ is a circle bundle with euler number e = 0, then

H 1 (N, Z) ∼ = Z 2g ⊕ Z/e
where g denotes the genus of Σ. In particular this implies that the absolute value of e is an invariant of the total space and not just the circle bundle. That in turn means that M and M are not homeomorphic.

Finally, we have to deal with the trivial bundle, that is, M ∼ = Σ × S 1 . In this case e(π) = 0 and

H 1 (M ; Z) ∼ = Z 2g+1 .

Now surface and circle covers

Σ → Σ and S 1 → S 1 of the same degree induce two covers

M ∼ = Σ × S 1 → M and M ∼ = Σ × S 1 → M
of the same degree. If g > 1, then Σ has strictly greater genus than Σ, so M is not homeomorphic to M .

Sol manifolds

In this section, we prove the following proposition.

Proposition 8.1. Sol 3-manifolds are not exceptional.

Proof. Every orientable Sol manifold M is finitely covered by a 2-torus bundle over S 1 with Anosov monodromy ϕ ∈ MCG(T 2 ) ∼ = SL 2 (Z) (see for instance [AFW15, Theorem 1.8.2]). Recall this the monodromy is called Anosov if the top-eigenvalue λ ϕ of ϕ as an SL 2 (Z)-matrix satisfies |λ ϕ | > 1. By Lemma 3.3, we may assume M is a 2-torus bundle over S 1 with Anosov monodromy ϕ. Let λ ϕ be the leading eigenvalue.

First we remind the reader of the well known fact that, as opposed to the case of hyperbolic mapping tori, the modulus of the eigenvalue λ ϕ is a topological invariant. Indeed, we have b 1 (M ) = 1. Each fibration π : M → S 1 with connected fibers and monodromy ψ induces a primitive non-torsion cohomology class [ψ] ∈ H 1 (M ; Z) and conversely each such cohomology class determines the fibration up to isotopy. The latter fact implies that the top eigenvalue λ ψ of the monodromy ψ depends only on [ψ]. The former observation implies that the only fibered classes are [ϕ] and -

[ϕ]. Since λ [-ϕ] = λ [ϕ]
,

|λ ϕ | is indeed a topological invariant
In order to build two non-homeomorphic covers, we proceed as follows. First let

T 2 → T 2
be a finite non-trivial characteristic cover. This means that ϕ lifts to a map

ϕ : T 2 → T 2
such that λ ϕ = λ ϕ . We obtain a cover

M → M,
where M ∼ = T 2 × [0, 1]/(x, 0) ∼ ( ϕ(x), 1).

Since b 1 (M ) = 1, we can also take a finite cyclic cover M → M of the same degree, say d = 1. The monodromy ϕ of this cover satisfies

λ ϕ = |λ ϕ | d ,
thus M and M are not homeomorphic.

Manifolds with non-trivial JSJ decompositions and non-trivial boundary

In this section, we prove the following proposition.

Proposition 9.1. Let M be an orientable, irreducible 3-manifold with empty or toroidal boundary such that either M has a non-trivial JSJ decomposition or ∂M is non-empty. Assume that M is not homeomorphic to S 1 × D 2 , T 2 × I or the twisted I-bundle over the Klein bottle, and that M is not a Sol manifold. Then M is not exceptional.

We start out with the following useful lemma.

Lemma 9.2. Let M be an orientable, irreducible 3-manifold with empty or toroidal boundary and let N be a JSJ component of M . Then for any finite group G and any surjective homomorphism f : π 1 (N ) G, there exist finite groups K and H and homomorphisms g, g 1 , g 2 , g 3 (of the type shown in the diagram), such that the following diagram commutes.

π 1 (N ) f i * / / g 3 $ $ $ $ π 1 (M ) g # # # # G K g 1 o o o o g 2 / / H.
Note in particular that the cover of N induced by the map g 3 is a cover of the one induced by f . Proof. For closed manifolds this lemma is an immediate consequence of [WZ10, Theorem A]. As is explained in [AFW15, (C.35)], the statement also holds in the case that M has non-empty toroidal boundary.

Proof of Proposition 9.1. In this proof we use the following terminology. Given a 3-manifold W with empty or toroidal boundary we refer to the union of the JSJ tori and the boundary tori of W as the set of characteristic tori of W . At the end of the upcoming proof we will have constructed two index d covering spaces M and M of M that we will distinguish by showing that they have unequal numbers of characteristic tori.

We say that a 3-manifold is tiny if it is homeomorphic to S 1 × D 2 , T 2 × I, or to the twisted I-bundle over the Klein bottle. Throughout this proof we will use on several occasions the following preliminary remark: If W is an orientable 3-manifold that is not tiny, then it follows from the classification of 3-manifolds with virtually solvable fundamental group, see [AFW15, Theorem 1.11.1], that no finite cover of W is tiny.

By [START_REF] Hempel | Residual finiteness for 3-manifolds[END_REF] (see also [AFW15, C.10]), we know that M has a finite-sheeted cover M → M such that each Seifert fibered JSJ component of M is an S 1 -bundle over a compact orientable surface. Since M is not a Sol manifold, neither is M . By our preliminary remark, since M is not tiny, neither is M . These three latter facts, along with [AFW15, Propositions 1.9.2 and 1.9.3], imply that this manifold M has the following useful property ( †): For any finite cover M → M , the preimage of the JSJ decomposition of M is exactly the JSJ decomposition of M .

Let N be a JSJ component of M , where possibly N = M . By hypothesis, ∂N is nonempty. There exists a finite-sheeted cover N → N such that the rank of the cokernel of the map H 1 (∂N ) → H 1 (N ) is at least one, by [AFW15, C.15, C.17]. (Here we used that N is not homeomorphic to T 2 × I, this follows from the fact that M is not tiny and from our hypothesis that M is not a Sol-manifold and from [AFW15, Proposition 1.6.2(3), 1.8.1, 1.10.1].)

The finite-sheeted cover N → N corresponds to a finite index subgroup of π 1 (N ). Recall that any finite index subgroup of a group contains a finite index normal subgroup, called its normal core; let the (finite index, regular) cover corresponding to the latter normal subgroup of π 1 (N ) be denoted N → N . By construction, N → N corresponds to the kernel of a surjective map π 1 (N )

G for some finite group G, and by Lemma 9.2, we obtain the following commutative diagram:

π 1 (N ) f i * / / g 3 ' ' ' ' π 1 (M ) g " " " " G ∼ = π 1 (N )/π 1 ( N ) K g 1 o o o o g 2 / / H.
Let M * be the cover M * → M corresponding to the kernel of g. From Lemma 9.2, it follows that the induced cover of N corresponding to g 3 is a finite-sheeted cover of N ; call it N * . Since M * is a finite-sheeted cover of M it follows from ( †) that N * is a JSJ component of M * . Since the cover N * → N is finite-sheeted it follows from an elementary argument, see [AFW15, A.12], that the rank of the cokernel of the map H 1 (∂N * ) → H 1 (N * ) is also at least one. Since M * is a finite-sheeted cover of M it follows from Lemma 3.3 that it suffices to show that M * is not exceptional.

We have the following commutative diagram, where the horizontal sequences form the long exact sequence in singular homology for the pairs (N * , ∂N * ) and (M * , M * \ Int(N * )) and the vertical arrows are induced by inclusion.

H 1 (∂N * ) H 1 (N * ) H 1 (N * , ∂N * ) H 1 (M * \ Int(N * )) H 1 (M * ) H 1 (M * , M * \ Int(N * )). i * k * j *
We obtain an induced commutative diagram

Coker(i * ) H 1 (N * , ∂N * ) Coker(j * ) H 1 (M * , M * \ Int(N * )). k *
By a standard excision argument, we see that H 1 (M * , M * \ Int(N * )) ∼ = H 1 (N * , ∂N * ); in other words, the rightmost vertical map above is an isomorphism. We see that k * is injective, and thus, the rank of Coker(j * ) is bounded below by the rank of Coker(i * ) which by hypothesis is at least one. Thus, there is an epimorphism Coker(j * ) Z. We can then define, for any m > 1,

f m : π 1 (M * ) H 1 (M * ) Coker(j * ) Z Z/m,

Ab

where the second and last maps are the canonical projections. Note that each characteristic torus of M * is contained in M * \ Int(N * ), and thus, by our construction, the image under inclusion of the fundamental group of any characteristic torus of M * lies in the kernel of f m for all m.

We are finally ready to construct two non-homeomorphic covers of M * with the same index. As mentioned in the beginning of the proof, we will do so by constructing two finite covers of M * with the same degree, but different number of characteristic tori. At this point we would like to recall that it follows from ( †) for any finite cover W of M * the characteristic tori of W are the preimages of the characteristic tori of M * . Now let n be the total number of characteristic tori of M * . Let T be a characteristic torus of M * . Since M * is not tiny (and in particular, M * is not homeomorphic to S 1 × D 2 , and thus, the boundary tori of M * are π 1 -injective) we can view π 1 (T ) as a subgroup of π 1 (M ). Since π 1 (M * ) is residually finite (Theorem 3.1), there is some finite index normal subgroup J π 1 (M * ) such that π 1 (T ) is not contained in J. Let d > 1 be the index of J in π 1 (M * ), and let p : M → M * be the d-sheeted cover of M * corresponding to J. Since π 1 (T ) is not contained in J we see that the preimage of T has strictly fewer than d components. Since the preimage under p of the characteristic tori for M * gives the characteristic tori for M , we see that the number of characteristic tori in M is strictly less than d • n. In order to build a second cover, let p : M → M * denote the cover corresponding to the kernel of f d : π 1 (M * ) → Z/d constructed above. By construction, the index of p is d and the number of characteristic tori in M is d • n, since the image under inclusion of characteristic torus of M * lies in the kernel of f d . We see that M and M are index d covers of M * but have an unequal number of characteristic tori, and thus are non-homeomorphic. 10. Non-prime and non-orientable 3-manifolds 10.1. Prime non-orientable 3-manifolds. Little further work is needed to completely characterize exceptional prime non-orientable 3-manifolds with empty or toroidal boundary. Such a manifold is either the twisted S 2 -bundle over S 1 , or irreducible. We already saw that the former is exceptional (Proposition 3.2). For the latter case, note that if the orientable double cover of a non-orientable 3-manifold M is not exceptional, then M is not exceptional. By our previous work, we only need to consider the irreducible non-orientable 3-manifolds whose orientable double covers are S 1 × S 2 , the 3-torus T 3 , S 1 × D 2 , S 1 × S 1 × [0, 1]. Here we have used that closed non-orientable 3-manifolds have positive first Betti number. In particular, their fundamental groups are infinite [AFW15, (E.3)].

Note that if a boundary torus for a manifold M is compressible, M has an S 1 ×D 2 summand. If M is prime, M must be homeomorphic to S 1 × D 2 , which is of course not non-orientable. Thus, a prime non-orientable 3-manifold with toroidal boundary, must have incompressible boundary. By [Swa73, Lemma 2.1], the fundamental group of the orientable double cover of an irreducible non-orientable 3-manifold M with boundary is free, which would be the case if the double cover is S 1 × S 2 or S 1 × D 2 , if and only if M is a homotopy RP 2 × S 1 . Such a manifold has first homology group Z ⊕ Z/2. Thus, it has two double covers, only one of which is orientable. As a result, it is not exceptional.

It remains to consider the non-orientable 3-manifolds with empty or toroidal boundary, whose orientable double cover is T 3 or T 2 × I. First suppose that M is a 3-manifold that is covered by T 3 . It follows from [MS86, Theorem 2.1] that M is Euclidean. This case is then addressed by Proposition 5.2. Secondly, if M is covered by T 2 × I, then a doubling argument shows, using the above, that M is either homeomorphic to the Klein bottle times an interval or to the Möbius band times S 1 . In either case M admits two 2-fold coverings, only one of which is orientable. Thus M is not exceptional. Thus, we have established the following proposition.

Proposition 10.1. Let M be a prime, non-orientable 3-manifold with empty or toroidal boundary. Then M is exceptional if and only if it is homeomorphic to S 1 ×S 2 . 10.2. Non-prime 3-manifolds. Finally, we consider non-prime 3-manifolds. Recall that a prime decomposition for a 3-manifold M is said to be normal if there is no S 1 × S 2 factor when M is non-orientable. It is well-known that every 3-manifold has a unique normal prime decomposition (see, for instance [Hem04, Theorem 3.15 and 3.21]). Note that we do not assume that the manifold is closed or orientable.

Below we give a general structure for covering spaces of non-prime 3-manifolds.

Proposition 10.2. Let M ∼ = M 1 # • • • #M k be a normal prime decomposition for a 3-manifold M . Then if M is a cover of M with index d, then

M ∼ = M 11 # • • • # M 1i 1 # • • • # M k1 # • • • # M ki k # S,
where S denotes S 1 × S 2 or S 1 ×S 2 whenever M is orientable or non-orientable respectively, and M j := M j1 • • • M ji 1 is a cover of M j with index d. Moreover,

(k -1) • d = k j=1 i j -1 + .
In addition, any M of this form is a cover of M with index d.

Since any cover of a prime 3-manifold is prime, the above gives a prime decomposition of M when we ignore S 3 summands.

Proof. Let p : M → M be the cover. Write M as M 1 \ B 1 ∪ • • • ∪ M k \ B k where each B j ⊂ M j is an open ball. Then each restricted map p| p -1 (M j \B j ) : p -1 (M j \ B j ) → M j \ B j is a covering map. Gluing balls along the lifts of the connected sum spheres gives rise to the covers M j .

It is clear that the cover M is built from the collection M j by identifying spheres in pairs, arising as lifts of the connected sum spheres. When the sphere pairs are in distinct connected components, we obtain a connected sum. When they lie in the same connected component, we obtain an S summand. Note that S can be chosen to be S 1 ×S 2 whenever M is non-orientable since N #S 1 × S 2 ∼ = N #S 1 ×S 2 whenever N is a non-orientable 3-manifold. The relationship between k, d, , and i j follows from the construction.

When M is oriented, each M j inherits an orientation and it is easy to see that M is a connected sum of oriented manifolds.

For M of the given form, an index d covering map M → M can be constructed by gluing together the individual covering maps.

For non-prime 3-manifolds, we have the following result.

Proposition 10.3. Let M be a non-prime 3-manifold with empty or toroidal boundary. Then M is exceptional if and only if it is homeomorphic to k • S 1 × S 2 , for some k ≥ 2.

Proof. First, we show that k • S 1 × S 2 is exceptional. Note that the only cover of S 1 × S 2 is itself. Then, we see immediately from Proposition 10.2 that any degree d cover of k • S 1 × S 2 is homeomorphic to ((k -1)d + 1) • S 1 × S 2 . Next, we show that any manifold which is not of the form k • S 1 × S 2 is not exceptional. Let M ∼ = M 1 # • • • #M k be a normal prime decomposition for M . By hypothesis, k ≥ 2.

As a preliminary step, we observe that if M has a single prime summand which is not exceptional, it is itself not exceptional. Since both S 2 -bundles over S 1 are exceptional, such a prime summand must be irreducible. Without loss of generality, assume that M 1 is irreducible and not exceptional. Then, there exist non-homeomorphic covers M 1 and M 1 of M 1 , both with index d, for some 1 < d < ∞. Construct the covers

M ∼ = M 1 #d(M 2 # • • • #M k ) and M ∼ = M 1 #d(M 2 # • • • #M k ) of M .
Note that both have index d. Since M 1 and M 1 are both irreducible, they appear in the prime decomposition of M and M respectively. By the uniqueness of normal prime decompositions, M and M are not homeomorphic. Thus, we only need to consider the case where each M i is itself exceptional. First we consider the case where M is orientable. Since M is not of the form k • S 1 × S 2 , we can assume, without loss of generality, that M 1 be an exceptional manifold other than S 1 × S 2 , that there exists a cover M 1 → M 1 of index d 1 and a cover M k → M k of index d k , such that d 1 ≤ d k .

We now build two covers of M with index d k as follows. Let

M = d k (M 1 # • • • #M k-1 )# M k , and let M = M 1 #(d k -d 1 )M 1 #(d 1 -1)S 2 × S 1 #d k (M 2 # • • • M k-1 )# M k .
Suppose that M ∼ = M . Then we see that M 1 ∼ = S 1 × S 2 , which is a contradiction.

Next, consider the case when M is non-orientable. Suppose first that there is at least one irreducible prime summand in the given normal prime decomposition of M . Without loss of generality, we can assume that this is M 1 . Since no non-orientable irreducible 3-manifold is exceptional, we see that M 1 is orientable. Let N denote the manifold M 2 # • • • #M k . Since M ∼ = M 1 #N is non-orientable, we see that N is non-orientable. Let N be the orientable double cover of N and construct the orientable double cover 2M 1 # N of M . Since M 1 is irreducible, it is in particular not S 1 × S 2 . By our argument in the previous paragraph, 2M 1 # N has two non-homeomorphic covers of the same index, showing that M is not exceptional.

It only remains to consider the case where M is non-orientable but has no irreducible prime summands. Since we have a normal prime decomposition, this implies M is of the form k•S 1 ×S 2 where k ≥ 2. Note that in this case H 1 (M ) ∼ = Z k where k ≥ 2. Then, we see that there are 2 k -1 connected double covers, only one of which is orientable, which completes the proof.

We observe that we have established the following proposition.

Proposition 10.4. The only non-orientable exceptional 3-manifold with empty or toroidal boundary is the twisted S 2 -bundle over S 1 .

Figure 1 .

 1 Figure 1. Leitfaden

  (g) := inf x∈X d X (x, gx) is positive. Then, denoting by Γ ah the set of such elements in Γ, we have: (6) sys(M ) = min γ∈Γ ah (γ).

  [Γ:Γ m ] and Γ m are both of index [Γ : Γ m ] and not isomorphic to each other. In fact a much stronger quantitative result holds. The strongest result (3), together with an argument due to Lubotzky and Belolipetsky-Gelander-Lubotzky-Shalev ([BGLM02], [BGLS10, Section 5.2]) shows that the number e d (M ) of pairwise non-isometric covers of a hyperbolic manifold of finite volume satisfies lim sup d→+∞ log e d (M ) d log(d) > 0. 4.3. Regular covers of hyperbolic manifolds. In this subsection we use (2), with r = 2, to prove the following result about regular covers. Proposition 4.5. Let Γ be the fundamental group of a hyperbolic 3-manifold (complete of finite volume). Then there exists sequences c n , d n → +∞ such that for each n we can find at least c n normal subgroups of index d n in Γ, which are pairwise non-isomorphic.

  g. [DKL12, Proposition 7.12]) and the fact that b n

  Suppose the induced circle and surface covers S 1 → S 1 and Σ → Σ have degrees m and respectively, then m = d and e( π) = m e(π).

This can be proved using the fact that all other RAAGs surject onto Z

* Z.

The criterion[START_REF] Margulis | Discrete subgroups of semisimple Lie groups[END_REF] Theorem (B) in Chapter IX] states that Γ has finite index in its commensurator Ω; since any lattice commensurable to Γ commensurates Γ, it has to be contained in Ω and the claim follows.

Email address: sfriedl@gmail.com URL: http://www.uni-regensburg.de/Fakultaeten/nat Fak I/friedl/ School of Mathematics, Georgia Institute of Technology, Atlanta, GA, USA Email address: junghwan.park@math.gatech.edu URL: http://people.math.gatech.edu/~jpark929/ Institut de Mathématiques de Jussieu -Paris Rive Gauche, Sorbonne Université, Paris, France Email address: bram.petri@imj-prg.fr URL: http://webusers.imj-prg.fr/~bram.petri/ Institut de Mathématiques de Toulouse, UMR 5219, Université Paul Sabatier-CNRS, 31062 Toulouse Cedex 9, France Email address: Jean.Raimbault@math.univ-toulouse.fr URL: https://www.math.univ-toulouse.fr/~jraimbau/ Max-Planck-Institut für Mathematik, Bonn, Germany Email address: aruray@mpim-bonn.mpg.de URL: http://people.mpim-bonn.mpg.de/aruray/