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Analytic, Reidemeister and homological torsion for congruence three--manifolds

Starting from the results in math.DG:1212.3161 we prove that for a given Bianchi group, certain natural coefficent modules and a lot of sequences of congruence subgroups of the size of the torsion subgroup of the first homology grows exponentially with the index (we give an explicit rate). We also prove limit multiplicity results for the irreducible components of the space of cuspidal forms.

1. Introduction 1.1. Torsion in the homology of arithmetic groups. The characteristic 0 homology of the arithmetic-and in particular congruence-lattices in real Lie groups is a well-studied invariant, interest in which has been prompted by sources ranging from three-manifold topology (Thurston's conjectures on hyperbolic three-manifolds) to number theory (as part of the Langlands programme), as well as its own sake. The torsion part of the integral homology has recently been at the center of a surge of interest, and the general problem this work is concerned with is the asymptotic behaviour of torsion homology in sequences of congruence lattices in a given Lie group, and more precisely in the growth of its size against the volume of the locally symmetric space associated to a lattice. For a sequence of congruence subgroups in a given lattice this behaviour is conjecturally described by N. Bergeron and A. Venkatesh in [4,Conjecture 1.3]. In the present paper we will deal only with hyperbolic three-manifolds (the locally symmetric spaces associated to the Lie group SL 2 (C) or SO (3,1)) where this conjecture is extended by the following question.
Question A. Let Γ be an arithmetic lattice in SL 2 (C) and Γ n a sequence of congruence subgroups of Γ. Let ρ, V be a finite-dimensional representation of SL 2 (C) and suppose there is a lattice V Z in V which is preserved by ρ(Γ). Does the limit lim sup n→∞ log |H 1 (Γ n ; V Z ) tors | [Γ : Γ n ] = −t (2) hold? Here t (2) (V ) < 0 is a constant that depends only on ρ; for ρ trivial it equals (6π) −1 , for ρ the tautological representation on C 2 it equals 11/(12π). This is still wide open for trivial coefficients; however there is a certain amount of computational evidence for a positive answer for certain particular sequences, see the tables in M.H. Şengun's [25] and the graphs in Section 4 of J. Brock and N. Dunfield's [6]. For some coefficient systemsincluding the tautological representation-the limit is proved to hold in [4] for a cocompact lattice Γ and an exhaustive sequence (meaning that every nontrivial element belongs only to a finite number of the Γ n -equivalently the injectivity radius of the M n = Γ n \H 3 goes to infinity). In the cocompact case at least one can reasonably promote A from a question to a conjecture (that the answer is affirmative), as it is proved (for the same twisted coefficients as those considered by [4]) by M. Abert, N. Bergeron, I. Biringer, T. Gelander, N. Nikolov, I. Samet and the author in [1], see also [22,Section 6.1].
The main goal of this paper is to give positive answers to Question A for the same local systems of coefficients as in [4] but for a non-cocompact Γ. More precisely we will consider the case where Γ is a Bianchi group, i.e. there is an imaginary quadratic field F such that Γ = SL 2 (O F ); it is wellknown that these groups represent all commensurability classes of arithmetic nonuniform lattices in SL 2 (C). However we do not manage to deal with all sequences of congruence subgroups of such a Γ (see 1.4.1 below) and we do not adress here the question of dealing with more general sequences of commensurable congruence groups. Note that the mere exponential growth of torsion-that is, the fact that the limit inferior of the sequence log |H 1 |/ vol be positive-is already known for certain sequences by work of the author [22,Section 6.5] and independent work of J. Pfaff [20]; the method used in the present paper is different from those in these two references, which in fact cannot yield such a precise result as the one we obtain: Theorem A. Let Γ be a Bianchi group, Γ n a cusp-uniform sequence of torsion-free congruence subgroups and M n = Γ n \H 3 . Let ρ, V be a real representation of SL 2 (C) which is strongly acyclic and V Z a lattice in V preserved by Γ. If V is strongly acyclic then we have Strong acyclicity of representations was introduced in [4], it means that the Hodge Laplace operators with coefficients in the local system induced by ρ have a uniform spectral gap for all hyperbolic manifolds; it was shown there to hold for all representations that are not fixed by the Cartan involution of SL 2 (C), in particular its nontrivial complex representations. Cusp-uniformity means that the cross-sections of all cusps of all M n form a relatively compact subset of the moduli space PSL 2 (Z)\H 2 of Euclidean tori. There are obvious sequences of congruence covers which are not cusp-uniform. The proofs of [21] actually apply not only to cusp-uniform sequences but to all BS-convergent sequences which satisfy a less restrictive condition on the geometry of their cusps, (1.3) below. However even this more relaxed hypothesis fails for some congruence sequences, see 1.4.1 below, and this raises the question of whether Question A actually has an affirmative answer in these cases. Examples of sequences to which our result does apply include the following congruence subgroups, which are all cusp-uniform: where M Y n n are obtained from M n by cutting off the cusps 1 along horospheres at a certain height Y n . The first task we need to complete is to check that the hypotheses of these theorems are satisfied by cusp-uniform congruence sequences. Regarding the BS-convergence we prove a quantitative result valid for any sequence of congruence subgroups (Theorem 3.1 below).
Theorem B. Let Γ n be a sequence of congruence of a Bianchi group; then the manifolds M n = Γ n \H 3 are BS-convergent to H 3 and we have in fact that there exists a δ > 0 such that for all R > 0 vol(M n ) ≤R ≤ C(vol M n ) 1−δ where C depends on Γ, R.
We note that this result is much simpler to prove in the cusp-uniform case (see 3.1.1). For the estimates on intertwining operators as well our proof goes well for any sequence of congruence subgroups. It is when we study the trace formula along the sequence M n that our arguments go awry, as some summands of the geometric side seem to diverge as n → ∞.
Then we want to use (1.2) to study cohomological torsion. The Reidemeister torsion τ abs (M Y n n ; V ) is related to the torsion in H 2 (M n ; V Z ), in fact it is defined as is the covolume of the lattice H p (M n ; V Z ) free in the space of harmonic forms satisfying absolute conditions on the boundary of M Y n n . The second thing to be done is to relate these to terms defined on the manifolds M n , using the description of H * (M n ; V C ) by non-cuspidal automorphic forms (namely, harmonic Eisenstein series). We then get a limit lim n→∞ log τ (M n ; V ) vol M n = t (2) (V ) (see (5.6) for the precise definition of τ ) and it remains to show that all terms in τ (M n ; V ) disappear in the limit except for |H 2 (M n ; V Z ) tors . This is done using elementary manipulations with the long exact sequence for the Borel-Serre compactification M n and its boundary, and estimates for the integrality of Eisenstein cohomology classes.

Limit multiplicities.
Another problem about sequences of congruence groups is the question of limit multiplicities for unitary representations of SL 2 (C). For such a representation π on a Hilbert space H π and a lattice Γ in SL 2 (C) one defines its multiplicity m(π, Γ) to be the largest integer m such that there is a SL 2 (C)-equivariant embedding of H m π into L 2 (Γ\SL 2 (C)). The question of limit multiplicities is then to determine the limit of the sequence m(π, Γ n )/ vol(Γ n \H 3 ) as Γ n ranges over the congruence subgroups of some arithmetic lattice. This question is of particular interest when π is a discrete series (when the limit is expected to be positive), and it has been considered in the uniform case by D.L. De George and N.R. Wallach in [10], by G. Savin [24] in the nonuniform case. In the case we consider there are no discrete series and thus we expect that the limit multiplicity of any representation will be 0.
A more precise question to ask is the following: the set G of irreducible unitary representations of G = SL 2 (C) is endowed with a Borel measure ν G (the Plancherel measure of Harish-Chandra), and for each lattice Γ ⊂ G the multiplicities in L 2 (Γ\G) define an atomic measure ν Γ . For a congruence sequence Γ n and a Borel set A ⊂ G, do we have ν Γn (A) ∼ vol(Γ n \H 3 )ν G (A) as n tends to infinity? In the case where the Γ n are congruence subgroups of a cocompact lattice this is shown to hold in [1,Section 7]. The non-compact case is much harder, but T. Finis, E. Lapid et W. Müller manage in [12] to deal with principal congruence subgroups in all groups GL n /E, E a number field. Here we will, more modestly, deal only with SL 2 over an imaginary quadratic field (but the sequences of congruence subgroups we consider are more general).
Theorem C. Let S be a regular Borel set in the unitary dual of G = SL 2 (C), Γ a Bianchi group and Γ n a cusp-uniform sequence of congruence subgroups. Then The question of limit multiplicities is related to the growth of Betti numbers in sequences of congruence subgroups via the Hodge-de Rham theorem. The latter has been studied in greater generality (for sequences of finite covers of finite CW-complexes) by W. Lück in [15] and M. Farber in [11]. Theorem 0.3 of the latter paper together with Theorem 1.12 of [1] imply that in a sequence of congruence subgroups of an arithmetic lattice the Betti numbers are sublinear in the volume in all degrees except possibly in the middle one where the growth is linear in the volume if the group has discrete series. Another proof of this for cocompact lattices, which actually yields explicit sublinear bounds in certain degrees, is also given in [1,Section 7]. For non-compact hyperbolic three-manifolds we dealt with this problem in [21]; a corollary of [21, Proposition C] and of Theorem B is then: Corollary D. Let Γ n be a sequence of torsion-free congruence subgroups of a Bianchi group Γ, then we have: Note that one of the proofs given in [21] is actually a very short and easy argument if one admits [1, Theorem 1.8].
1.4.1. Sequences that are not cusp-uniform. As noted there the results of [21] are valid under a slightly less restrictive condition than cusp-uniformity: it suffices that we have for some α > 0, where Λ n,j are the Euclidean lattices associated to the h n cusps of M n . This is clearly implied by cusp-uniformity in view of Lemma 3.2, and implies the unipotent part of BSconvergence. It is not hard to see that there are exemples of congruence sequence which satisfy this condition but are not cusp-uniform. However, there are congruence sequences which do not satisfy (1.3), for example those associated to the subgroups K n f which are the preimage in in this case there are n cusps having α 1 ≍ 1 and α 2 ≍ n and the index is about n 3 . However I have no clue as to whether the limit multiplicities and approximation result should or not be valid for these sequences.

1.4.2.
Orbifolds. Theorems C and B are valid for sequences of orbifolds as well (see [22]). We have not included the necessary additions here in order to keep this paper to a reasonable length and because they are quite straightforward. The approximation for analytic torsion carries to this setting as well, though the Cheeger-Müller equality for manifolds with boundary which is one of the main ingredients in the proof of Theorem 5 has not, to the best of my knowledge, been proven for orbifolds yet. As for the final steps of the proof of Theorem A they either remain identical (if the cuspidal subgroups are torsion-free) or are simplified by the presence of finite stabilizers for the cusps, which may kill the homology and the continuous spectrum. We will not adress this here, some details are given in [22].

Trivial coefficients.
For the topologist or the group theorist the trivial local system is the most natural and interesting. The approximation of analytic L 2 -torsion [21,Theorem A], [4,Theorem 4.5] extends to that setting if one assumes that the small eigenvalues on forms have a distribution which is uniformly similar to that of the L 2 -eigenvalues on H 3 . Let us describe more precisely what this means. Let M n be a sequence of congruence covers of some arithmetic three-manifold, we know by Theorem C that for p = 0, 1 the number m p ([0, δ]; M n ) = ( λ∈[0,δ] m p (λ; M n ) of eigenvalues of the Laplace operator on p-forms on M n in an interval [0, δ] behaves asymptotically as m p is the pushforward of the Plancherel measure. We would need to know that we have in fact a uniform decay of m p ([0, δ]; M n )/ vol M n as δ → 0, for example for all δ > 0 small enough and some absolute C, c. We will now describe a (very) idealized situation in which (1.4) would hold in a particularly nice form. Let α p ∈]0, ∞ + ] be the pth Novikov-Shubin invariant of H 3 (see [16,Chapters 2 and 5]) then there would be an absolute constant C > 0 such that for any δ > 0 small enough and any congruence hyperbolic three-manifold M n we have For functions we have α 0 = ∞ + (meaning there is a spectral gap on H 3 ) and (1.4) is known to hold in this case, and in fact in a much more general situation, by L. Clozel's solution of the "Conjecture τ " [8]. For 1-forms α 1 = 1 and one should probably not expect to prove (1.5) literally (or even for it to hold in this form). We ask the following question, which to the best of our knowledge is wide open.
Question B. Does there exists λ 0 > 0 such that for any ε > 0 there is a C ε > 0 such that for any congruence hyperbolic three-manifold M and δ ≤ λ 0 we have (Or less precisely, does this hold for some exponent c > 0 in place of 1 + ε on the right-hand side?) A positive answer to this question is not enough to imply a positive answer to Question A as one still has to analyze the "regulator" terms in the Reidemeister torsion: see [4, 9.1]. In some cases this is dealt with in a work in progress by N. Bergeron, M.H. Şengun and A. Venkatesh.

1.4.4.
Non-arithmetic manifolds. If Γ is a non-arithmetic lattice in SL 2 (C) then it is conjugated into SL 2 (O E [a −1 ]) for some number field E and algebraic integer a. Thus we can define its congruence covers (whose level will be coprime to a), and it can be proved (see 3.1.1) that they are BSconvergent to H 3 . Question A still makes sense for trivial coefficients, but is not expected to always have a positive answer in that setting. It is actually expected that for some sequences of nonarithmetic covers with inj M n → +∞ the order of the torsion part of H 1 (M n ; Z) does not satisfy (1.6). We record the following less precise questions: • Let M be a finite-volume hyperbolic three-manifold, does there exists a sequence M n of finite covers of M with inj M n → +∞ and (1.6) lim n→+∞ log |H 1 (M n ; Z) tors vol M n = 1 6π ?
(a stronger form of this was dubbed "Topological Volume Conjecture" by T.T.Q. Le).

5
• Let M n be a sequence of finite-volume hyperbolic three-manifolds such that M n BSconverges to H 3 and their Cheeger constants are uniformly bounded from below. Do we have Here T (M n ) is the Ray-Singer analytic torsion, regularized as in [18] if the M n have cusps. For compact manifolds this is Conjecture 1.4 in [1], and for covers [6,Conjecture 1.13].
We refer to [4, 9.1] and [6] for more complete discussion around these questions.
1.5. Outline. Section 2 introduces the background we use throughout the paper. In Section 3 we prove Theorem B, and in section 4 we estimate the norm of intertwining operators, thus completing the proof of Theorem C and of (1.2). Section 5 completes the proof of the asymptotic Cheeger-Müller equality between analytic and Reidemeister torsions of the manifolds M n . The final section 6 analyses the individual behaviour of the terms in the Reidemeister torsion, finishing the proof of Theorem A.
1.6. Acknowledgments. This paper originates from my Ph.D. thesis [22], which was done under the supervision of Nicolas Bergeron, and it is a pleasure to thank him again for his guidance. The results here are significantly better than those I obtained previously, and the final version of this paper was mostly worked out during a stay at Stanford University; I am grateful to this institution for its hospitality and to Akshay Venkatesh for inviting me and for various discussions around the topic of this paper.

Notation
2.1. Bianchi groups and congruence manifolds. For this section we fix an imaginary quadratic field F and let Γ = SL 2 (O F ) be the associated Bianchi group. We will denote by A f the ring of finite adèles of F . At infinity we fix the maximal compact subgroup K ∞ of SL 2 (C) to be SU(2); if K ′ f is a compact-open subgroup of SL 2 (A f ) we will adopt the convention of denoting by K ′ the compact-open subgroup K ∞ K ′ f of SL 2 (A).
2.1.1. Congruence subgroups. For any finite place v of F let K v be the closure of Γ in SL 2 (F v ) ; Let Γ(I) be defined by (1.1) and K f (I) its closure in K f ; then Γ(I) = Γ ∩ K f (I) so that Γ(I) is indeed a congruence subgroup ; likewise, Γ 0 and Γ 1 (I) are "congruence-closed", i.e. they are equal to the intersection of their closure with SL 2 (F ).
For a compact-open subgroup K ′ f ⊂ K f we will denote by Γ K ′ = SL 2 (F ) ∩ K ′ f the associated congruence lattice ; we define its level to be the largest I such that K f (I) ⊂ K ′ f 2 . Then the following fact is well-known (see [22,Lemma 5.8]).
where I is the level of K ′ f .
2 This is well-defined as K f (I)K f (I ′ ) = K f (J) where J = gcd(I, I ′ ). 6 2.1.2. Congruence manifolds. For a compact-open K ′ f we denote by M K ′ the orbifold Γ K ′ \H 3 . The strong approximation theorem for SL 2 (which in this case is a rather direct consequence of the Chinese remainder theorem) the subgroup SL 2 (F ) is dense in SL 2 (A f ), and it follows that we have an homeomorphism 2.1.3. Unipotent subgroups. Let N be a unipotent subgroup of SL 2 defined over F , B its normalizer in SL 2 and α the morphism from B/N to the multiplicative group 3 given by the conjugacy action on N. For any place v of F we have the Iwasawa decomposition and this yields also that SL 2 (A) = B(A)K. We define a height function on SL 2 (A) by this does not depend on the F -rational unipotent subgroup N. We fix the unipotent subgroup 0 N to be the stabilizer of the point (0, 1) in affine 2-space, and we identify it with the additive group using the isomorphism ψ sending 1 to the matrix 1 1 0 1 . Then N is conjugated to 0 N by some g ∈ SL 2 (F ) and we identify N with the additive group using the isomorphism given by the composition of conjugation by g with ψ•α(b) −1 where g = bk (and we view α(b) as an automorphism of the additive group).
In particular, since the ideals of O F are a uniform family of lattices it follows easily that the families Γ(I), Γ 0 (I) or Γ 1 (I) are cusp-uniform (see [ On a proper quotient we always take the pushforward measure, in particular the measure on SL 2 (A) is the pushforward of dbdk in the Iwasawa decomposition SL 2 (A) = B(A)K.

Spaces of functions.
Let A 1 be the subgroup of T(A) such that every character from T(A) to R × + factors through A 1 ; then T(F ) ∼ = F × is contained in A 1 and we will denote by H the Hilbert space L 2 (F × \(A 1 K)). We have a natural isomorphism where C(F ) is the class-group of F . Let χ be a Hecke character, then it induces characters of A 1 and B(A) that we will continue to denote by χ. For s ∈ C and φ ∈ H ∩ C ∞ (A 1 K) there is a unique extension φ s of φ to SL 2 (A) which satisfies: We will denote by H s the space of such extensions, and H s (χ) its subspace of functions having A 1 -type χ on the left. The space H s is acted upon by SL 2 (A) by right translation; when s = 0, 1 the decomposition of H s into SL 2 (A)-irreducible factors is given by the H s (χ). If τ is a finitedimensional complex continuous representation of K we define H s (χ, τ ) to be the subspace of H s (χ) containing the functions which have K-type τ on the right (in other words, the projection to H s (χ) of the subspace of K-invariant vectors in H s (χ) ⊗ τ ).

The height function y defined by (2.2) is left B(F )N(A)-invariant, and it is well-known (cf. [22, Lemme 5.23]) that the series
converges absolutely for all g ∈ SL 2 (A) and Re(s) > 2, uniformly on compact sets. For φ ∈ H the function φ s defined through (2.3) we denote E(φ s ) = E(φ, s) which is convergent for Re(s) > 2 according to the above. We have the following fundamental result, due to Selberg for SL 2 /Q and to Langlands in all generality (see [22, 5.4.1] for a simpler proof in this case, based on ideas of Godement).
Proposition 2.2. The function s → E(φ, s) has a meromorphic continuation to C, which is homolorphic everywhere if χ = 1. If χ = 1 there is only one pole of order one at s = 1.
The main point of the theory of Eisenstein series is that they give the orthogonal complement to the discrete part of the regular representation on L 2 (SL 2 (F )\SL 2 (A). Define L 2 cusp (SL 2 (F )\SL 2 (A)) to be the closed subspace of all functions on SL 2 (F )\SL 2 (A) whose constant term (defined by (2.5) below) vanishes.

2.2.4.
Intertwining operators. Let f be a continuous function on SL 2 (F )\SL 2 (A). We define its constant term to be Let φ ∈ H, φ s be defined by (2.3) and f = E(φ s ). We use the Bruhat decomposition and when Re(s) > 3/2 we get: we define the intertwining operator Ψ(s) on H by so that we obtain (using the notation of (2.3)): . For Re(s) = 1 2 the map Ψ(s) is an isometry for the inner product of H.

2.2.5.
Maass-Selberg. Finally we record the Maass-Selberg expansions [22, 5.4 When s ∈ R this degenerates to 2.3.1. Differential forms and L 2 (SL 2 (F )\SL 2 (A)). Let K ′ be a compact-open subgroup of K, and ρ a representation of SL 2 (C) on a finite-dimensional real vector space V . We can associate to ρ a local system on M = M K ′ and we denote by Ω p , L 2 Ω p (M ; V ) the spaces of smooth and squareintegrable p-forms on M with coefficients in V (see [21, 2.3]). It is well-known that there is an identification Let τ be the representation of K who has for it's finite part C[K/K ′ ] and whose infinite part is equal to the representation of K ∞ on V C ⊗ ∧ p p * . We define the map E(s, .) to be such that the following diagram commutes: where h is the number of cusps of M and we identify (2.12) where: ) under the identifications on the right side of (2.11); • W l is a certain subspace of V C , and we have V C = l W l . For more details see [21, Section 3.1]. We will skip the definition for 1-forms since it is basically the same.

Homology and cohomology.
Here we consider a CW-complex X, Λ = π 1 (X) and L a free Z-module of finite rank with a Λ-action. There are then defined chain an cochain complexes If Y is a sub-CW-complex of X and i its inclusion in X we have the following property In case there is a perfect duality between two Λ-modules L, L ′ which comes from a Λ-invariant bilinear form we get a Kronecker pairing on H p (X; L) free × H p (X; L ′ ) free satisfying (2.13); for Γ a lattice in SL 2 (C) there exists such a self-duality for the Γ-modules V = V n 1 ,n 2 (given by the form induced on V by the determinant pairing on V ). This form bilinear form, which we will denote by ·, · V , is actually defined over Z[m −1 ] for m = n 1 !n 2 !, and if V Z ⊂ V Q is a lattice we put We suppose now that X is a n-dimensional compact manifold with boundary ∂X. Poincaré duality is an isomorphism of graded Z-modules H * (X; V ) . It is compatible with the long exact sequences of the pair X, ∂X in the following sense [5, Theorem V.9.3].
Proposition 2.4. The diagram: is commutative, where horizontal lines are the long exact sequences in homology and cohomology of X, ∂X and vertical arrows are Poincaré duality morphisms.

Asymptotic geometry of congruence manifolds and approximation of L 2 -invariants
In this section we shall, assuming the results of the next section, prove Theorem C from the introduction and the approximation result for analytic torsion (Theorem 3.8 below).
Theorem 3.1. There is a δ > 0 such that for any Bianchi group Γ = Γ(O F ) and sequence Γ n of torsion-free congruence subgroups in Γ, for all R > 0 there is a C > 0 such that In particular, the sequence of hyperbolic manifolds M n = Γ n \H 3 is BS-convergent to H 3 .
We record the following much weaker consequence (see Lemma 3.3 below) of this as a separate fact; note that this is actually the only part of Theorem 3.1 that we make full use of here, and a direct proof is much easier than that of the latter. Recall that K f is the closure in SL 2 (A f ) of Γ = SL 2 (O F ) and let K ′ f be a closed finite-index subgroup with level I. We will show that for the subgroup Γ 3.1.1. Remarks. 1. For cusp-uniform sequences Theorem 3.1 follows immediately from (the proof of) Proposition 4.7 in [22] and [1,Theorem 1.11]. In general, the mere BS-convergence (without the precise estimates) follows from the latter with a little more work: one can see that it implies that any invariant random subgroup which is a limit of a sequence of congruence covers has to be supported on unipotent subgroups, which is impossible (for example because of "Borel's density theorem" [1, Theorem 2.6]).
2. The proofs below and in [1] actually yield a stronger result: that there are ε, δ > 0 such that Benjamini-Schramm convergence of manifolds with cusps. We recall some notation: for a hyperbolic manifold M we let N R (M ) be the number of closed geodesics of length less than R on M . If Λ is a lattice in C we define Then the ratio α 2 /α 1 only depends on the conformal class of Λ, in particular if Γ ∋ −1 is a lattice in SL 2 (C) and N a unipotent subgroup such that Γ ∩ N is nontrivial (we will say that N is Γ-rational) then In particular to prove that a sequence of finite covers of a fixed orbifold is BS-convergent we need only give o(vol M n )-bounds for the right-hand side.
3.1.3. Proof of Theorem 3.1, the loxodromic part. Here we recall how the bound on N R follows from the results in [1,Section 5]. Let c be a closed geodesic in the orbifold Γ\H 3 and γ ∈ Γ any element of the associated loxodromic conjugacy class in Γ. For any g ∈ K f we have that γ fixes the coset gK ′ f if and only if gγg −1 belongs to K ′ f , so that the number of lifts of c in M K ′ is equal to the number of fixed points of γ in K f /K ′ f . By Theorem 1.11 in [1] there are constants δ (depending on F ) and C (depending on c) such that the latter is less than This shows that for a given R there is a C R such that for all K ′ we have where the j N are representatives of the Γ K ′ -conjugacy classes of unipotent subgroups in SL 2 (F ). We fix a unipotent subgroup N in SL 2 (F ), and let N = N(A f ) ∩ K f ; clearly it suffices to prove that (3.1) holds if we sum only over the unipotent groups contained in the Γ-conjugacy class of N(F ). These Γ K ′ -conjugacy classes are in natural bijection with the set of double cosets N \K f /K ′ f For p ∈ Z a rational prime we will denote F p = Q p ⊗ Q F and K p the closure of Γ in SL 2 (F p ). The latter is isomorphic to We will denote by K p (p k ) the compact-open subgroup of matrices congruent to 1 modulo p k , and by g p the Lie algebra of K p . The crucial case is when we have K ′ p = K p for all but one rational prime p. We identify N(F p ) with F p (see 2.1.3 above) and for a finite-index subgroup L of N(F p ) we put with absolute constants, so that we must bound the sum .
We rewrite S p as follows: we fix a k ≥ 1 such that where the second line follows from the equality . On the other hand B p normalizes N p and we can quotient on the right to get: which is the sum that we will now estimate. For l = 0, . . . , k − 1 we define We may suppose that K p (p k−1 ) ⊂ K ′ p , and we will prove in 3.1.5 below the following estimate for d l when l ≤ k/3: It follows that on the other hand we have [K p : K ′ p ] ≤ 2p 6k and we finally get Now we return to the general case ; let m be an integer such that K f (m) ⊂ K ′ f , as above we have that We can rewrite this as where the second inequality follows from (3.5). It follows from [1,Lemma 5.9] that there are constants c, C ≥ 1 such that for any compact-open subgroup K ′ f ⊂ K f , if K ′ p is its projection to K p then we have where the product runs over all rational primes such that K ′ p = K p , so that we get which finishes the proof of (3.1) (we get δ = 1 54c ).
3.1.5. Proof of (3.4). This proof is reminescent of that of [1, Proposition 5.11], albeit much more cumbersome due to the fact that we cannot identify the precise elements of N p which are conjugated into K ′ p . Under the hypothesis that K ′ p ⊃ K p (p k−1 ), for any l = 1, . . . , k − 1 we have that K ′ p K p (p l+1 )/K p (p l+1 ) cannot contain a generating set for the 6-dimensional F p -Lie algebra g p /pg p = K p (p l )/K p (p l+1 ). For a subset Y ⊂ K p define We will prove the following lemma at the end of the section.
Lemma 3.4. If p is unramified and p = 2, 3 then a proper subgroup of K p /K p (p) (resp. a proper Lie subalgebra of g p /pg p ) cannot contain more than p + 1 pairwise noncommuting unipotent (resp. nilpotent) elements.
This implies that q X l (0) ≤ p + 1 for all l ≤ k/3 (since there are only finitely ramified primes we get q X l (0) ≤ Cp for a C > 0 depending on F , we will work with C = 1 to simplify notation). Now we deal with j ≥ 1: we will prove that when j < (k − 2l)/3 we must have q X l (j) ≤ p, which in view of (3.6) implies immediately (3.4) for l ≤ k/3. Suppose that there is an h ∈ K p /K p (p j ) such that conjugating K ′ p by h we may suppose that h = 1. This means that there exists pairwise distinct Computing the right-hand side yields that Now the worst that can happen is that we are (up to conjugation) in at most one of the following situations: We may suppose that all t m = 1; now let i, i ′ such that a = c i + c i ′ ∈ F p , modulo p 2l+j+1 we have which is not in 1+ p 2l+j V so that we see that K ′ p contains K p (p 2(l+j) ). In case (c) we can do exactly the same reasoning to get that K ′ p ⊃ K p (p 2l+3j ). It remains to deal with case (b), which is again similar: we have that K ′ p K p (p l+2j+1 ) contains 1 + p l+2j V ′ , and if we supppose that all t m c m = 1 and t i + t i ′ ∈ F p we get that modulo p 2l+j , and multiplying by some other g m s to kill the top-right coefficients we get that 1 + p 2l+j u ∈ K ′ p K p (p 2l+j+1 ) for some u ∈ V ′ , which shows vthat K ′ p contains K p (p 2(l+j) ) also in this case.
In conclusion, we have seen that if q X l (j) > p then K ′ p contains K p (p 2l+3j ) which implies that j ≥ (k − 2l)/3, which finishes the proof of (3.4).
3.1.6. Proof of Lemma 3.4. It follows from the following classification as the image X of any of the proper subgroups listed here contains less than p + 1 unipotent, pairwise noncummuting elements.
Lemma 3.5. Let H be a subgroup of K p /K p (p) such that H contains two unipotent elements which do not commute ; then There is a similar statement for proper Lie subalgebras of g p .
In case p = 3 is inert this follows immediately from Dickson's Theorem [13,Theorem 8.4 in Chapter 2] (this is where we use p = 3). In the remaining case where p is split the lemma is actually much simpler, since the projection of H on one of the factors SL 2 (F p ) must contain two non-commuting unipotent elements and we can then apply Dickson's theorem (which in this case is almost trivial).
The result for Lie algebras is easier in the inert case: if a subalgebra contains two noncommuting nilpotent elements then their Lie bracket is an element in the Cartan subalgebra contained in the intersection of their normalizers. If this bracket is not F p -rational then we get the whole algebra since its adjoint action on each of the nilpotent F p 2 -subalgebras is irreducible (here we use p = 2), if it is then they generate a subalgebra conjugated to sl 2 (F p ). The ramified case is dealt with as for the case of groups.
3.2. Limit multiplicities. We will prove in 3.2.3 below that Theorem C follows from the following less precise result. Let ρ, V be a finite-dimensional representation of SL 2 (C) and for an hyperbolic orbifold  3.2.1. Regularized trace. The first step towards Theorem 3.6 is to prove the convergence of regularized traces; the following result is an immediate consequence of Theorem 4.5 in [21] and Theorem 3.1 above.
Proposition 3.7. Let Γ n be a sequence of cusp-uniform congruence subgroups of a given Bianchi group. Then we have the limit for any φ ∈ A(R).
Recall from loc. cit. that A(R) is a C ∞ -dense subset of the Schwartz functions on R whose Fourier transforms yield point-pair invariants of rapid decay on H 3 × H 3 .

3.2.2.
Proof of Theorem 3.6. Because A(R) is dense in L 2 (R) it suffices (by approximating the characteristic function of regular sets) to prove that for any φ ∈ A(R) we have that Tr φ(∆ p cusp [M n ])/ vol M n converges to Tr (2) be the restriction of ∆ p [M n ] to the subspace of cusp forms. By standard arguments one deduces Theorem 3.6 from the fact that for all φ in a L 1 -dense subset of C ∞ c (R) we have According to Proposition 3.7 this would follow if we can prove We will show this when p = 0. From (2.12) we get: (where the summand 0/1 comes from the subspace of harmonic sections, equals 0 when V is acyclic). As Ψ(iu) is unitary for u ∈ R we obtain Putting and applying Proposition 4.1 we get for any ε > 0 the bound where the integral on the right-hand side converges absolutely since C ε (u) is polynomially bounded. By Lemma 3.2 we get that for ε small enough it is in fact an o(vol M n ), which finishes the proof of (3.8) and of the theorem.

Laplacian eigenvalues and representations.
The fact that we can deduce limit multiplicities for representations (Theorem C) from limit multiplicities for Laplacian eigenvalues (Theorem 3.6) is a consequence of the fact, which is specific to real-rank-one groups, that a unitary representation of SL 2 (C) is determined by its Casimir eigenvalue and its SU(2)-types. More precisely, the unitary representations of SL 2 (C) are parametrized by (Z × iR)/ ∼ ∪]0, 2[ where (l, ia) ∼ (−l, −ia). On the other hand the SU(2)-types are the restrictions to SU(2) of the holomorphic representations V n = V n,0 of SL 2 (C), and by Frobenius reciprocity the representations containing the SU(2)-type V n are the π ±l,ia for 0 ≤ l ≤ n and n − l = 0 (mod 2), so we can deduce Theorem C by an easy induction on l using the limit multiplicities for m 0 V l and m 1 V l .

Approximation for analytic torsion.
Theorem 3.8. Let Γ n be a cusp-uniform sequence of torsion-free congruence subgroups of Γ M n = Γ n \H 3 , then we have Proof. According to Theorem A in [21] we have to check two conditions: • The sequence M n is BS-convergent to H 3 • There is an ε > 0 such that there exists a C > 0 so that for all u ∈ [−ε, ε] we have The BS-convergence is the content of Theorem 3.1 above. To prove (3.9) note that we have for all u ∈ R the bound (where the second majoration follows from Lemma 2.1), and taking ε = δ/6 shows that the righthand side of (3.10) is indeed an o(vol M n ), uniformly for u in a compact set.

Estimates on the logarithmic derivatives of intertwining operators
This section is devoted to the proof of the following result. , τ = τ ∞ ⊗ τ f and φ ∈ H(χ, τ ). Let ε > 0; there exists a polynomially bounded function C ε on R depending only on F, ε and τ ∞ such that for all u ∈ R.
We will suppose that φ ∈ C ∞ (K) is equal to a product ⊗ v φ v ; then it is enough to show that Since H s (χ) is an irreducible unitary representation of SL 2 (A) when Re(s) = 1/2 and Ψ(s) −1 d du Ψ(s) is a SL 2 (A)-equivariant endomorphism of H s (χ) it is a scalar operator, say c Id. Now Ψ(s) −1 is unitary and thus for any two φ, φ ′ ∈ H s (χ) we have d du Ψ(s) H = d du Ψ(s) H , so that it suffices to consider a single function φ ∈ H s (χ); we will take φ v to be the spherical vector at unramified places and specify φ v for each ramified place; the infinite place does not matter very much for our purposes here.

Computation of the intertwining integrals.
Let v be a finite place; we will make repeated use of the matrix decomposition: We now compute the L 2 -norms of I v and J v for the function φ v defined as follows: We compute this for k = w a b c d : and ax + c is nonzero for all x ∈ O v ); and it follows immediately that and it follows that and since χ is nontrivial on 1 + π mv −m v O v the integral on the right vanishes.

Unramified places.
If χ is not ramified at v then we choose φ v to be the function in H s which is identically equal to 1 on K v . We recall the following well-knwown computation:

Infinite place. We finally compute
For z ∈ C we have and it follows that As |z| → +∞ we have k z − z/|z| z/|z| ≪ |z| −1 2 ∞ (for any norm |.| on K ∞ ). For Re(s) > 1/2 we get: where the O(1) depends on τ ∞ but not on s. If χ ∞ = 1 then the integral is zero so that I ∞ is bounded independantly of s for Re(s) ≥ 1/2. If χ ∞ = 1 we have and it follows that I ∞ has a meromorphic continuation to Re s > 0 such that I ∞ (k)−π Γ(2s−1) Γ(2s) φ ∞ (k) is bounded independantly of s for Re(s) ≥ 1/2. We will write s = σ + iu for this whole subsection and suppose (unless otherwise stated) that σ = 1/2. Taking the derivative of the product (4.5) yields so that we get, using the functional equation for L(χ, .): We will suppose at first that χ is non-trivial so that the L-function L(χ, .) is holomorphic on Re(s) > 0. To bound both L 1 and L 2 we use the following well-known lemma.  Proof. The Euler product for L(χ, 2s) yields for Re(s) > 1/2 the absolutely converging series expansion

The series
converges absolutely for Re(s) > 1/4 and its sum is bounded by a constant depending only on F , so that we are left with estimating v ∈Sχ (log q v )q −2s v χ(π v ). Let χ 1 , . . . , χ h F be all the Hecke characters on F × \A 1 such that ker(χ j ) ⊃ M . If χ is any Hecke character there exists some j such that χ(π v ) = χ j (π v ) for all places v ∈ S χ . We then have that v ∈Sχ As χ j is non-trivial the function H : and as q ≤ |f χ | we are left with d du L(χ, 2s) L(χ, 2s) ≤ C(u)(log |f χ |) 2 where C(u) has polynomial growth in u, which finishes the proof of the lemma.
Since d du I ∞ γ(s) is also bounded by log |f χ | we thus get where C(u) is growing polynomially. It remains to deal with the case where χ = 1, which is the same except that we have to group the terms I ∞ and L(χ, 2s − 1) = ζ F (2s − 1) to cancel their poles at s = 1/2. The details will be left to the reader (see also [22, 5.5.3]).

4.3.
Estimates off the critical line. The estimates above are still valid for any real number s (except if χ is trivial and s = 1) but different exponents for |I| and f χ are obtained. We will not require tight estimates for those, and be content with stating the following rough result.
where C depends on F, ε, σ and τ ∞ and c only on σ.

Reidemeister torsion and asymptotic Cheeger-Müller equality
The aim of this section is to define a Reidemeister torsion τ for congruence manifolds with cusps and then prove the following result. We fix a Bianchi group Γ and a strongly acyclic Γ-module V Z .
Theorem 5.1. Let Γ n be a cusp-uniform sequence of pairwise distinct torsion-free congruence subgroups of Γ. Let M n = Γ n \H 3 and let τ (M n ; V Z ) be defined by (5.6) below, then we have We now explain how this result follows from [21] and the results in 5.4 and 5.5 below. According to Theorem 3.1 and the proof of Theorem 3.8 we can apply Theorem B in [21] to the sequence M n , so that we get for the sequence Y n described there. According to Proposition 5.7 below we can replace Y n by any Υ n such that max Υ n j ≤ |I n | c for some constant c, and the result now follows from Proposition 5.4 below.
Before giving the definition of τ (M ; V Z ) and the proof of Proposition 5.4 we will recall from scratch how to describe analytically the cohomology of the boundary ∂M of the Borel-Serre compactification M of an hyperbolic manifold with cusps, and how to construct a section of the pull-back map H * (M ) → H * (∂M ) using Eisenstein series as in [14] (see also [3, Section 3]). 5.1. Boundary cohomology. We fix a congruence 5 manifold M = M K ′ ; the set of its cusps is in bijection with C( Recall from [21] that W l,k is the subspace of V C of weight (l, k) for the (complex) representation of sl 2 (C) ⊗ C = sl 2 (C) ⊕ sl 2 (C). In degree 1 we have an isomorphism 5 Everything in the next three sections applies to all finite-volume hyperbolic three-manifolds. 22 defined as follows: to a 2h-tuple of vectors v 1 , . . . , v h ∈ W −n 1 ,n 2 ,v 1 , . . . ,v h ∈ V n 1 ,−n 2 we associate the de Rham cohomology class [ω] of the 1-form ω given by Let us check that ω is indeed a closed form. We have v j = w j ⊗ u j where w j = g j λ j e 0 and u j = g j e n 2 , so that ρ(g j n z g −1 j ).v j = w j ⊗ ( n 2 l=0 Q l (z)g j e n 1 −l where Q l is a polynomial depending only on n 2 . It follows that z → ρ(n z )v j is anti-holomorphic. We can see in the same way that z → ρ(n z )v j is holomorphic and all this yields that We will denote by H 1,0 (∂M ; V C ), H 0,1 (∂M ; V C ) the subspaces of H 1 corresponding respectively to C[C(K ′ )] ⊗ W ∓n 1 ,±n 2 .
As for degrees 0 and 2 we have isomorphisms Indeed, the space W n 1 ,n 2 is the space of fixed vectors of 0 N in V , and to v 1 , . . . , v h ∈ V n 1 −n 2 we associate the holomorphic section h j=1 g j v j or the holomorphic 2-form h j=1 (dz j ∧ dz j ) ⊗ (g j v j )). Now we will give an explicit description of the maps i * p following [14]. For a closed p-form f ∈ Ω p (M ; V C ) we denote by [f ] its de Rham cohomology class. Given an harmonic form ω ∈ H 1 (∂M ; V C ) and a s ∈ C we can form the Eisenstein series E(s, ω) ∈ Ω 1 (M ; V C ). The following result is well-known, see for instance the proof of [14, Theorem 2].
Proof. We need only check that if P is a Γ-rational parabolic subgroup the constant term of E(s 1 V , ω) at P is a closed form on Γ P \H 3 . It is equal (in the SL 2 (C)-equivariant model for E ρ , see [21, (2.5)]) to ω +Φ + (s 1 V )ω and as ω, and thus the second statement follows. The constant term of E(s 1 V , ω) is also not an exact form since its restriction to ∂M is not, and it follows that the map H 1,0 (∂M ; , ω] is injective. As we have the equality of dimensions dim H 1,0 (∂M ; We can thus define an application E 1 : for ω ∈ H 1,0 (∂M ; V C ), ω ∈ H 0,1 (∂M ; V C ). From the formula for the constant term of Eisenstein series we get 23 and it follows that In degree 2 the long exact sequence shows that i * 2 is onto (since H 3 (M , ∂M ; V C ) ∼ = H 0 (M ; V C ) = 0). We have a result akin to Lemma 5.2 for this case, whose proof is very similar.
Computing the Casimir eigenvalue (cf. [21, (2.4) ] is nonzero, and by equality of dimensions we get that these classes span H 2 (M ; V C ).
We denote by E 2 the map H 0 (∂M ;

5.3.
Inner products on cohomology and Reidemeister torsion. From now on we will suppose that V = V n 1 ,n 2 with n 1 > n 2 , so that s 1 v ≥ 1. It follows from the Maass-Selberg relations (2.9) that for ω ∈ H 1,0 (∂M ; V C ) we have the limit and we define an inner product on H 1 Similarly, we can put

(5.5)
Now for p = 1, 2 the integral cohomology H p (M ; V Z ) free is a lattice in the hermitian vector space H p (M ; V C ). We finally define the Reidemeister torsion of M with coefficients in V by the formula

Asymptotic equality of Reidemeister torsions.
We prove now that the Reidemeister torsion we just defined is asymptotically equal to the absolute Reidemeister torsion of the truncated manifolds (for a certain choice of truncations).
Proposition 5.4. Let Γ n , V be as in the statement of Theorem 5.1. There exists a sequence Υ n such that and max j Υ n j ≤ |I n | c for some c > 0. The first step is the following result, whose proof is essentially contained in [7, 6.8.3].
Lemma 5.5. There are C, c > 0 depending only on F such that the following holds. Let Γ ′ ⊂ Γ be a congruence subgroup, M = Γ ′ \H 3 , h its number of cusps, α j = α 1 (Λ n,j ) where Λ n,j , j = 1, . . . , h are the euclidean lattices corresponding to the cusps of M ′ . Then for all Y ∈ [1, +∞) h such that for . Now we check that f ′ Y satisfies absolute boundary conditions: close enough to the boundary we have f ′ Y = f P , and since dy ∧ * f P = 0 and df P = 0 we conclude that f ′ Y ∈ Ω 1 abs (M Y ; V C ). Thus, we have According to Proposition 6.8 in [21], up to making C larger we may suppose that for Y j ≥ Cα j the Laplace operator ∆ 1 abs [M Y ] has no eigenvalue in the open interval ]0, λ 1 [ (for some λ 1 > 0 depending only on V ) as soon as Y j ≥ Cα j , and we then get from (5.8) and (5.9) that where the last line is a consequence of Cauchy-Schwarz inequality.
Proof of Proposition 5.4. Let Υ ∈ [1, +∞) hn ; we have and we will thus show that for p = 1, 2 we have We will deal only with p = 1, the case p = 2 being similar. Let r n be the restriction map H 1 (M n ; V C ) → H 1 (M Υ n ; V C ). As the inclusion M Υ n ⊂ M n is an homotopy equivalence, it induces an isomorphism between the cohomology groups and we get that vol H 1 (M Υ n ; V Z ) free = | det(r n )| vol H 1 (M n ; V Z ) free where the determinant is taken with respect to unitary bases on each space (the left-hand space being endowed with the inner product defined by (5.4) and the right-hand on with the L 2 inner product coming from harmonic forms). We will show below that log | det(r n )| = o(vol M n ), in fact that |r n |, |r n | −1 ≤ 1 + ε n for some sequence ε n such that b 1 (M n ; V C ) log ε n = o(vol M n )). We take back the notation f Υ from Lemma 5.5, if f is a closed form on M n we have r n where the second inequality follows from Lemma 5.5 and the rough bound vol(M Υ − M Υ/2 ) ≤ C hn j=1 α 2 (Λ n,j ) α 1 (Λ n,j ) . Now we will bound the right-hand side using the following lemma.
Lemma 5.6. Let Y = max j Υ j , a = s 1 V and I n be the level of Γ n . There are b, C > 0 (depending on F and V ) such that . The Maass-Selberg relations (2.9) yield that

From proposition 4.3 it now follows that
which deals with the upper bound; the lower bound is proved in a similar manner.
We have hn j=1 α 2 (Λ n,j ) and it now follows from (5.10) and Lemma 5.6 that for some e > 0 we have The lower bound for f Υ L 2 (M Υ ) is more subtle. We have where the second minoration follows from Lemma 5.5. We have max α j n ≪ |I| 1 2 and also vol(M Υ − M Υ/2 ) ≪ j α 2 α 1 which is bounded by |I n | 3 , and it follows from Lemma 5.6 that For A large enough and Υ n j = |I n | A−1 we get from (5.12) and (5.13) that Thus |r n | −1 ≤ 2 and |r n | ≤ C|I n | Ae and as dim H 1 (M n ; V C ) = h n it follows that | log det(r n )| ≪ h n log |I n |, and as h n ≪ (vol M n ) 1−δ (Lemma 3.2) the right-hand side is an o(vol M n ), as we wanted to show.

5.5.
Comparing absolute torsions. The following result is necessary to be able to use together Proposition 5.4 below and Theorem B in [21], and its proof completes that of Theorem 5.1.
Proposition 5.7. Let Y n be the sequence from Theorem B of [21]. For any sequence Υ n ∈ [1, +∞) hn such that there is a c > 0 for which Y n j ≤ Υ n j ≤ |I| c we have Proof. It suffices to prove the result for Υ n j = |I n | c . We will use a smooth family of Riemannian metrics g u , u ∈ [1, +∞) on M such that Let us prove that such a family exists. We identify a collar neighbourhood N of the boundary in M with j T j × [0, 1] where the T j are the boundary components of ∂M 1 . The metrics g u defined as follows do the job, as can be checked by an easy computation: on M − N ∼ = M 1 g u is the hyperbolic metric, an in the cusps we put 1] where h is a bump function which takes the value 1 for t ≤ 0 and 1 for t ≥ 1 and we identify V ⊥ (x,t) = T x T j and V (x,t) with the orthogonal complement of the latter in T x M 1 . Let * u be the Hodge star for g u , put Then we have [23,Theorem 7.6] d du log τ abs (M ; g u ) = tr(α u ).
Let λ u be the largest eigenvalue of α u , so that Thus the result would follow if we proved that |λ u | ≪ 1 u for u ≥ Y j . First we compute the eigenvalues: if f is an eigenform of α u with norm 1 and eigenvalue λ we have Indeed, * −1 • * f, f = λ, so that we get λ = M Now let f be a harmonic 1-form for the metric g u which is an eigenform for α u ; we want to see that d|f | gu /du ≪ u −1 . On M − M 1 write f = f 1 + f 2 according to the decompostion T M = V ⊕ V ⊥ (in coordinates f 1 is the composant on dy), then according to (5.14) we have the pointwise inequality The fact that f is co-closed implies that f 1 has a vanishing constant term and it follows that where the estimate is a consequence of [7, Lemma 6.2.1]. The right-hand side is ≪ u −1 : indeed, the sequence Y n j was defined in [21] as vol M n hn j=1 (α 2 (Λ n,j )/α 1 (Λ n,j )) 2 1 10 and it follows from Lemma 3.2 and the cusp-uniformity of the M n that Y j ≫ α 1 (λ j )|I n | δ for some δ > 0. Thus, as we consider only |I n | c ≥ u ≥ Y n j /2 we get u α 1 (Λ j ) ≫ u η for some η > 0 (depending on Υ n ) and clearly e −u η ≪ u −1 .

Torsion in (co)homology
We can now finish the proof of Theorem A, whose statement we recall below. Theorem 6.1. Let Γ be a Bianchi group, Γ n a cusp-uniform sequence of torsion-free congruence subgroups and M n = Γ n \H 3 . Let V be a real representation of SL 2 (C) and V Z a lattice in V preserved by Γ. If V is strongly acyclic then we have Let us describe how the results in this section articulate to yield this result. Recall that in (5.6) we have defined a Reidemeister torsion for the congruence manifolds M n = Γ n \H 3 the logarithm of whose is given by and by Lemmas 6.8, 6.10 and 6.9 below the only term in (6.3) that is not o(vol M n ) as n → ∞ on the left is − log |H 2 (M n ; V Z ) tors |. This proves (6.2); we will deduce (6.1) from it in 6.5 at the end of the section.
6.1. Integral homology of the boundary. We have previously described the cohomology of the boundary with coefficients in V C using differential forms; to analyze the terms (6.3) we will need a precise description of the integral homology and cohomology through cell complexes.
6.1.1. Cell complexes. Let T be a 2-tori, U a finite-rank free Z-module with a representation ρ : π 1 (T ) → SL(U ). We fix a cell structure on T with one 2-cell, e 2 , two 1-cells e 1 1 , e 1 2 and one 0-cell e 0 and denote by u 1 , u 2 the associated basis for π 1 (T ) (i.e. u i is the homotopy class of the loop e 1 i ). Then we have an isomorphism of Z-complexes C * ( T ; U ) ∼ = C * ( T ) ⊗ U which yields an isomorphism of graduated modules In this model the differentials for C * (T ; U ) are given by Proof. We prove the result only for Λ = 1 + O F X ∞ (where X ∞ = 0 1 0 0 ), the general case can be reduced to that particular one. In the proof of Lemma 6.5 below we will show that if 1 + aX ∞ ∈ Λ ′ then (Λ ′ − 1)V Z ⊃ N aV Z , where V Z = ker ρ(X ∞ ). In particular, putting d = dim V we get: and the result about H 0 follows at once. Write now Λ ′ = Zu 1 ⊕ Zu 2 . From (6.4) we know that H 1 (Λ ′ ; V Z ) embeds in (V Z ⊕ V Z )/ im(ρ(u 1 )− 1) ⊕ (ρ(u 2 ) − 1). The Z-torsion of the latter itself embeds into Now this last module has a torsion the order of which is bounded by (N |u 1 | 2 ×N |u 2 | 2 ) d ≪ [Λ : Λ ′ ] 4d , and this finishes the proof for H 1 .
6.1.3. Free part of the homology. Suppose now that T has an Euclidean structure, so that its homology groups with coefficients in V C are endowed with the L 2 inner product and have a Hodge decomposition H 1 = H 1,0 ⊕ H 0,1 , which in the case of a boundary component of an hyperbolic manifold corresponds to the decomposition in 5.1. We use the rational structure on the F -vector space V Q = Sym n 1 F 2 ⊗ Sym n 2 F 2 given by restricting the scalars from F to Q, which induces a rational structure on H * (Γ; Lemma 6.3. The subspaces H 1,0 (T ; V C ) and H 0,1 (T ; V C ) are rational ; moreover there are constants C, c depending only on T, V such that for any finite cover T ′ of T of degree D, we have Proof. Let z be a complex coordinate for T ′ and put ω 1 = dz ⊗ (ρ(n z )e 0,n 2 ), ω 2 = dz ⊗ (ρ(n z )e n 1 ,0 ), then ω 1 , ω 2 are generators (over C) for H 1,0 (T ′ ; V C ) and H 0,1 (T ′ ; V C ) respectively; we will check that they are rational and that they generate (over O F ) a lattice whose index in H 1 is bounded by a polynomial in D.
Let us show how to proceed to the construction of (6.5): write ρ(z)e n 1 −l,0 = e n 1 −l,0 + ; for ease of notation we identify v = e 0 ⊗ v, then we get that Iterating this procedure until we reach n 1 we get a cycle whose coefficients are rational fractions in u 1 , u 2 , so that we only have to multiply by its denominator to get (6.5).
The index of the lattice O F θ 1 ⊕ O F θ 2 in H 1 (T ′ ; V Z ) is bounded by a polynomial in u 1 , u 2 , which is itself bounded by a polynomial in D, say R(D). On the other hand the computations (6.6), (6.7) show that N R(D)ω 2 , M R(D)ω 1 are integral, and clearly the index of the lattice they span is bounded by a polynomial in u 1 , u 2 , which finishes the proof.
We now easily deduce : Proof. For p = 0, 2 this is trivial. For p = (1, 0), (0, 1) we see that the L 2 -norms of the integral classes constructed in the proof of Lemma 6.3 are polynomially bounded in vol T ′ , which implies the result in this case, and for p = 1 we deduce it from the latter and Lemma 6.3.
6.2. Subexponential growth of torsion. We prove here that in degrees other than 2 the torsion in cohomology has subexponential growth. 6.2.1. Homology in degree 0. Lemma 6.5. Let Γ n be a sequence of congruence subgroups in Γ(O F ), M n = Γ n \H 3 . We have that log |H 0 (M n ; V Z )| = o(vol M n ) Proof. We prove the result for principal congruence subgroups and then deduce the general case. To do the former we will show that N IV Z ⊂ (Γ(I) − 1)V Z for all I and some integer N depending only on n 1 , n 2 , so that |H 0 (M I ; V Z )| ≤ (N |I|) dim V from which it follows at once that log |H 0 (M I ; Let X ∞ = 0 1 0 0 and X 0 = 0 0 1 0 . For a ∈ I we have that η a = 1 + aX ∞ ∈ Γ(I).
We begin by studying the case where n 2 = 0, n 1 = n; put e 1 = (1, 0) and e 2 = (0, 1), then the family e n 1 , e n−1 1 e 2 , . . . , e n 2 is an O F -basis of the free module V Z . Let N be the product of all binomial coefficients n k , we will see that N ae n−k 6.2.2. Cohomology in degree 1. We will use the following elementary lemma in what follows. Lemma 6.6. Let A ∈ Hom(Z m , Z n ) and B ∈ Hom(Z n , Z m ) such that for all ϕ ∈ Hom(Z m , Z) and v ∈ Z n we have (ϕ, Bv) = (ϕ • A, v). Then Z m /BZ n and Z n /AZ m have the same torsion subgroup.
Proof. In appropriate bases of Z m , Z n the matrices of A and B are transpose of each other. Lemma 6.7. We have Proof. Recall that there is a Γ(O F )-invariant pairing on V Q and let V ′ Z is the lattice in V which is dual to V Z through this pairing. For notational ease we will use H * , H * to denote (co)homology with coefficients in V Z and H ′ * , H * ′ for V ′ Z -coefficients. The existence of the Kronecker pairing and the property (2.13), together with Lemma 6.6 imply that and it further follows that where the equality follows from Poincaré duality and the majoration from the segment H ′ 1 (M , ∂M ) ≤ log |H ′ 0 (M ) tors | + log |H 1 (∂M ) tors | where the inequality on the second line follows from (6.9). The right-hand side is an o(vol M n ), as follows from Lemmas 6.5 and 6.2, which finishes the proof.
6.3. Integrality of Eisenstein classes. We give bounds for the denominators of Eisenstein classes, which will be needed to control the volumes in degree 1. We use Q, Z to denote the algebraic closure of Q and its ring of integers. We will prove in this subsection very unprecise results which are sufficient for us; related computations are in [3] and [7, Chapter 6.7]. Proposition 6.8. There are constants C, c depending on F and V such that for any congruence subgroup Γ ′ ⊂ Γ of level I there is an integer N ∈ Z whose absolute norm N 0 ∈ Z is bounded by 1 ≤ N 0 ≤ C|I| c and such that for any ω ∈ H 1 (∂M ; V Z ) we have 6.3.1. Denominators of differential forms. We will denote M the hyperbolic manifold Γ ′ \SL 2 (C)/K ∞ and K ′ f the closure of Γ ′ in K f . Let f be a closed differential form in Ω 1 (M ; V C ) which represents a class in H 1 (M ; V Q ); then its denominator is the smallest a ∈ Z such that a [f ], c M ∈ Z for all c ∈ H 1 (M ; V ′ Z ) where V ′ Z is the dual lattice of V Z for the determinant pairing ·, · V on V Q . We want to express this in terms of values of functions in H for those classes which come from Eisenstein series.
The denominator of a class [f ] ∈ H 1 (M ; V ) is equal to the lcm of the denominators of the integrals i c i f, v i V where i c i ⊗ v i runs over the 1-cycles in C * (M ; V ′ Z ). Since all those cycles that are not torsion (in which case the integral is 0) come from the boundary as the cohomology H * cusp (M ; V ) is zero, we get that the denominator of [f ] is equal to that of its constant term, which we need only integrate against closed cycles on the boundary. The homotopy classes of loops on the component T j = Λ j \N j of ∂M are represented by closed 1-parameter subgroups of T j , which correspond to rational points n j,Q of the Lie algebra of N j .
Suppose for the moment that f ∈ Ω 1 (M ). Let X ∈ n j,Z be a primitive vector (i.e. not multiple of another integer vector) and X 0 ∈ n 0 , γ ∈ SL 2 (F ) such that X = Ad(γ)X 0 ; let g ∞ ∈ SL 2 (C) and let c be the simple cycle which is the image of c = {exp(tX)g, t ∈ R} in M . Letf be the function in (C ∞ (SL 2 (A)) ⊗ p * ) K ′ corresponding to f (see (2.11)), and φ 0 = f (·)(X 0 ) ∈ C ∞ (SL 2 (A)) K ′ f , then we have that where SL 2 (A f ) ∋ g f ∈ γK ′ f . Now if f ∈ Ω 1 (M ; V C ) and v ∈ V C we can apply the above argument to the 1-form ρ · f, v V and it follows that c ρ · f, v V = φ 0 (g ∞ g f ) as above.
If f is an Eisenstein series then φ 0 above belongs to the subspace C ∞ (N(A)B(F )\SL 2 (A)), and in fact equals φ s + (Ψ(s)φ) −s for some φ ∈ H. For a closed form E 1 (ω), ω ∈ H 1 (∂M ; V C ) we get φ 0 = φ + Ψ(s 1 V )φ (the integral against a cycle does not depend on the height in the cusp). Now if ω is integral, the function φ takes integral values when v ∈ V Z ; thus we have translated our original problem of estimating the denominator of E 1 (ω) for ω an harmonic 1-form on ∂M whose integrals against integral 1-cycles is integral into the problem of estimating the denominator of the values of Ψ(s 1 V )φ for φ ∈ H K ′ f taking integral values on K.
6.3.2. Proof of Proposition 6.8. We have seen above that the Proposition reduces to proving that if φ ∈ H K ′ f takes integral values then there is a N with absolute norm N 0 ≤ C|I| c such that N Ψ(s 1 V )φ takes integral values. We will prove this only for φ ∈ H(χ) K ′ f where χ is a Hecke character with conductor dividing I; the general result follows since these φ generate H K ′ f ; moreover we will suppose that K ′ f = K f (I) (which is also not a problem in view of Lemma 2.1). Let S I be ∞ together with all finite places such that K ′ f is not maximal at v, i.e. IO v = π mv v O v for a m v ≥ 1. Then we have seen that The values of χ v on O × v , π v are algebraic numbers whose denominators and numerators have absolute norms bounded by Cq b v where b, C depend only on χ ∞ and F , and it follows easily from the computations of the local integrals given in 4.1.1 (which until the last steps are valid for any φ v and not only the specific choice made there; see also [22, p. 145]) that we have for all k ∈ K v and some rational integers C, c depending only on F, V (where the quotient of local L-factors equals one when χ ramifies at v). It follows from(6.10) that The key ingredient now is the result of Damerell [9] about special values of Hecke L-functions, a corollary of which (as we explain in 6.3.3 below) is that there exists rational integers m, M ∈ Z and some number Ω ∈ C × depending on F, V such that the L-values L(χ, s 1 V − 1), L(χ, s 1 V ) belong to ΩπQ, ΩQ respectively and the denominators of L(χ, s 1 V − 1)/(Ωπ), L(χ, s 1 V )/Ω both divide M |f χ | m . This yields Ψ(s)φ(k) ∈ Ψ ∞ (φ ∞ )(k ∞ ) (πC|I| c ) −1 Z.
Now we know that Ψ(s) is defined over Q and it follows that Ψ ∞ (φ ∞ )(k ∞ ) ∈ πQ, and as the component at infinity lives in a finite-dimensional space that does not depend on I this finishes the proof 6 .

Damerell's theorem.
To state this result precisely we need some notation. Let E be the Hilbert class field of F : it is a finite abelian extension of F such that for every ideal I of F , IO E is a principal ideal of O E . We will denote m I a generator for this ideal (an "ideal number" of I in Damerell's paper). Damerell's result [9,Theorem 2] implies that there exists some Ω ∈ C × which depends only on F such that if we put γ = 6m One can likely compute the term at the infinite place more precisely to see directly that it belongs to πm −1 Z for some m ≥ 1 depending only on V .

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for an ideal I then for any Hecke character χ such that f χ divides I we have that (6.11) γ L(χ, s 1 V ) Ω 2s 1 V ∈ Z and 2γπ L(χ, s 1 The absolute norm of the ideal IO E equals |I| [E:F ] and we thus see that (6.11) implies the result stated in the proof of Proposition 6.8 above.
6.3.4. Remark. Note that the computations in [7, Section 6.7] allow to control the denominators of Eisenstein classes obtained from integral classes on the boundary also when the local system V has nontrivial L 2 -cohomology, so that Proposition 6.8 should also be valid in this case. Proof. We will deal with lim inf ≥ 0 and lim sup ≤ 0 separately; we shall begin by proving that (6.12) lim inf n→∞ log vol H 1 (M n ; V Z ) vol M n ≥ 0.
The embedding H 1 (M n ; V C ) → H 1 (∂M n ; V C ) is isometric by definition of the inner product on H 1 (M n ; V C ) and its image is the subspace {ω + Φ + (s 1 V )ω, ω ∈ H 1,0 (∂M n ; V C )}. Let π be the orthogonal projection of H 1 (∂M n ; V C ) onto H 1,0 (∂M n ; V C ). Then Lemma 6.3 implies that the image π(H 1 (∂M n ; V Z )) contains H 1,0 (∂M n ; V Z ) with an index which is ≪ (vol M n ) 4hn . As vol π(i * 1 H 1 (M n ; V Z )) ≤ vol i * 1 H 1 (M n ; V Z ) we get that vol H 1 (M n ; V Z ) ≫ (vol M n ) 4hn vol H 1,0 (∂M n ; V Z ) and together with Lemmas 6.4 and 3.2 this finishes to prove (6.12).
Let us now show that (6.13) lim sup n→∞ log vol H 1 (M n ; V Z ) vol M n ≤ 0.