Integrality of volumes of representations

Let M be an oriented complete hyperbolic n-manifold of finite volume. Using the definition of volume of a representation previously given by the authors in [3] we show that the volume of a representation ρ:π1(M)→Isom+(Hn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho :\pi _1(M)\rightarrow \mathrm {Isom}^+({{\mathbb {H}}}^n)$$\end{document}, properly normalized, takes integer values if n is even and ≥4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ge 4$$\end{document}. If M is not compact and 3-dimensional, it is known that the volume is not locally constant. In this case we give explicit examples of representations with volume as arbitrary as the volume of hyperbolic manifolds obtained from M via Dehn fillings.


Introduction
Given a representation ρ : → Isom + (H n ), our central object of study is the volume Vol(ρ) of ρ as defined in [6] for n = 2 and in general in [3]. This notion extends the classical one introduced in [15] for M compact and, as it was shown in [20], if M is of finite volume it coincides with definitions introduced by other authors [10,12,19].
We refer to Sect. 4.2 for the definition of Vol(ρ) and content ourselves with listing some of its main properties. (a) n = 2 and ρ is the holonomy representation of a (possibly infinite volume) complete hyperbolization of the smooth surface underlying M, [5,14,21], or (b) n = 3 and ρ is conjugate to Id , [3,4,10,11].
(3) The volume function is continuous on Hom( , Isom + (H n )) (see Proposition A. 1 in Appendix A). (4) If either M is compact and n ≥ 2 [29] (see also [9]) or M has finite volume and n ≥ 4 [20], the volume is constant on connected components of the representation variety. As a consequence it takes only finitely many values. where here and in the sequel, Vol(S n ) is the volume of the n-sphere S n of constant curvature 1.
Our main result is a generalization of the integrality property (6) to the case in which M is not compact, and n is even and ≥ 4. We remark that this is in sharp contrast with (5).
We recall that the Bieberbach number is the smallest integer B d such that any compact flat d-manifold has a covering of degree B d that is a torus. Such d-manifolds occur as connected components of the boundary of a compact core. Recall in fact that, in the context of hyperbolic geometry a compact core N of M is a compact submanifold that is obtained as the quotient by of the complement in H d+1 of a -invariant family of pairwise disjoint open horoballs centered at the cusps. The strategy of the proof of Theorem 1.1 builds on results in [6], where the authors studied the case in which dim M = 2 and established congruence relations for Vol(ρ). In order to implement this strategy, we show that the continuous bounded class defined by the volume form on H 2m , has a canonical representative in the bounded Borel cohomology of SO(2m, 1) • that, under the change of coefficients Z → R, corresponds to (−1) m 2 Vol(S 2m ) ω b 2m . For 2m = 2, ε b 2 coincides with the classical bounded Euler class as defined in [13]. Then we establish a congruence relation modulo Z for (−1) m 2 Vol(S 2m ) Vol(ρ) in terms of invariants attached to the boundary components of N that are now assumed to be (2m − 1)-tori. If T i is a component of ∂ N and ρ i : Z 2m−1 → SO(2m, 1) • is the restriction of ρ to π 1 (T i ) Z 2m−1 , then the invariant attached to T i is In the case in which m = 1, ρ * i (ε b 2 ) coincides with the negative of the rotation number of ρ * i (1) ∈ SO(2, 1) • and we show in § 5 that, if m ≥ 2, ρ * i (ε b 2m ) always vanishes. Remark In general has always a subgroup of finite index all whose cusps are toric. However little is known about which collections of compact flat (2m − 1)-manifolds N are the components of the boundary of a compact core as above. If dim M = 4, it is known that there are compact flat 3-manifolds that are not diffeomorphic to ∂ N , [23,Corollary 1.4], while on the positive side there are hyperbolic 4-manifolds with ∂ N that is a 3-torus, [22]. Which leads to the following: QUESTION If is the fundamental group of a compact flat (2m − 1)-manifold and ρ : → SO(2m, 1) • is any representation, does ρ * (ε b 2m ) vanish for 2m ≥ 4? Thus it is really in all odd dimensions that the nature of the values of Vol remain mysterious, though, according to the results in [20], for n ≥ 4 there are only finitely many possibilities.
We end this introduction by giving a result in dimension 3. In this case, the character variety of < Isom + (H 3 ) is smooth near Id , and its complex dimension near Id equals the number h of cusps of M [31]. As a consequence of the volume rigidity theorem and the continuity of Vol, the image of Vol contains at least an interval [Vol(M) − , Vol(M)] for some > 0. Special points in the character variety of come from Dehn fillings of M. Let M τ denote the compact manifold obtained from M by Dehn surgery along a choice of h simple closed loops τ = {τ 1 , . . . , τ h }. If the length of each geodesic loop τ j is larger than 2π , M τ admits a hyperbolic structure [31] and an analytic formula for Vol(M τ ) depending on the length of the τ j has been given in [27].
Thus with our cohomological definition, Vol(ρ) gives a continuous interpolation between the special values Vol(M τ ). A natural question here is whether Vol is real analytic.
The structure of the paper is as follows. In § 2 we summarize the main facts about various group cohomology theories used in this paper. In § 3 we define a Borel cohomology class ε 2m ∈ H 2m B (SO(2m, 1), Z ) with coefficients (see § 2) and relate it to an explicit multiple of the class ω 2m ∈ H 2m B (SO(2m, 1), R ) defined by the volume form on H 2m (see (3.1)). In § 4 we first show that ε 2m has a unique bounded representative ε b 2m ∈ H 2m B,b (SO(2m, 1), Z ) (Proposition 4.2); in § 4.2 we proceed to define the volume Vol(ρ) of a representation ρ : π 1 (M) → Isom + (H 2m ) = SO(2m, 1) • and use the bounded integral class ε b 2m to establish a congruence relation for Vol(ρ) (Theorem 4.4). In § 5, which is the core of the paper, we show that the contributions from the various boundary components of a compact core of M to the congruence relation all vanish for 2m ≥ 4. In § 6 we relate the volume of the representations of π 1 (M) obtained by Dehn surgery to the volumes of the corresponding manifolds. In the Appendix we prove the continuity of the map ρ → Vol(ρ).

Various cohomology theories
We collect in this section cohomological results that will be used throughout the paper.
Given a locally compact second countable group G, we consider Z, R and R/Z as trivial modules. If : G → {−1, +1} is a continuous homomorphism, we denote by Z and R the corresponding coefficient G-modules, where g * t = (g)t for t ∈ Z, R and by R /Z the corresponding quotient module.
If A is any of the above G-modules, H • B (G, A) denotes the cohomology of the complex of Borel measurable A-valued cochains on G. If A = R, R , we will also need the continuous cohomology H • c (G, A) with coefficients in A and we will use that for A = R, R the comparison map To compute the continuous cohomology of G we use repeatedly that if G × V → V is a proper smooth action of a Lie group G on a smooth manifold V , there is an isomorphism since every G-invariant differential form on G/K is closed. Another result we will use is Wigner's isomorphism [32], or rather a special case thereof [1, Theorem E]: namely if A = Z or A = Z , there is a natural isomorphism where BG is the classifying space of G and H • sing refers to singular cohomology. A vanishing theorem that is often used states that if L is compact, then as can be readily verified using the regularization operators defined in [7, § 4].
Analogously to the vanishing of the continuous cohomology for compact groups, if P is amenable, then for all k ≥ 1. Consider now the two short exact sequences of coefficients and Using that R → R/Z admits a bounded Borel section, one obtains readily, both for the trivial and the nontrivial modules, long exact sequences in Borel and in bounded Borel cohomology with commutative squares coming from the comparison maps c Z and c R between these two cohomology theories: and . . .
where δ and δ b are the connecting homomorphisms. realizes O(2m) as a maximal compact subgroup of both SO(2m, 1) and GL(2m, R) and induces homotopy equivalences

Proportionality between volume and Euler class
The homomorphism that to a matrix associates the sign of its determinant, coincides on O(2m) with the restriction of and will be denoted by . Thus we obtain isomorphisms The (universal) Euler class ε univ 2m ∈ H 2m sing (BGL + (2m, R), Z) is a singular class in the integral cohomology of the classifying space BGL + (2m, R) of oriented R 2mvector bundles (see [25, § 9]). It is the obstruction to the existence of a nowhere vanishing section. As it changes sign when the orientation is reversed, it extends to a class ε univ 2m ∈ H 2m sing (BGL(2m, R), Z ). Furthermore, if M is a closed oriented 2mdimensional manifold, its tangent bundle is classified by a (unique up to homotopy) classifying map inducing a map and thus and call Euler class the group cohomology class corresponding to ε univ 2m under the composition of the isomorphisms in (3.2) and (3.4).
, the image of ε 2m under the change of coefficients Z → R is a multiple of the volume class ω 2m . We show now that this multiple is given by Indeed, let M be a closed oriented hyperbolic manifold. The lattice embedding i : π 1 (M) → SO(2m, 1) induces classifying maps Extending the principal group structure from O(2m) to SO(2m, 1) gives the principal SO(2m, 1)-bundle The latter is isomorphic to which is the flat principal SO(2m, 1)-bundle associated to the lattice embedding i : π 1 (M) → SO(2m, 1). It follows that the composition from (3.6) classifies the tangent bundle T M. As a consequence, Thus, by the naturality of Wigner's isomorphism, the following diagram commutes: We deduce that Moreover, the hyperbolic volume of M is obviously given by The Chern-Gauss-Bonnet Theorem [30,Chapter 13,Theorem 26] implies that if M is a hyperbolic (2m)-dimensional manifolds then where the proportionality constant is up to the sign (−1) m the quotient between the Euler characteristic and the volume of the compact dual of hyperbolic space, namely the (2m)-sphere of constant curvature +1. Finally, since the lattice embedding induces an injection we obtain the value of the proportionality constant in (3.5).

Remark 3.1 Recall that the orientation cocycle
Or : assigns the value ±1 to distinct triple of points according to their orientation, and 0 otherwise. Identifying S 1 with ∂H 2 we obtain by evaluation a Borel cocycle and a cohomology class Since the area of ideal hyperbolic triangles in H 2 is ±π depending on the orientation, this cocycle represents (1/π )ω 2 = −2ε 2 , where the last equality comes from (3.5).

The bounded Euler class
In this section we recall that the volume class of hyperbolic n-space admits a unique bounded representative and establish for n even the analogous result for the Euler class, which leads to the definition of the bounded Euler class in Proposition 4.2. In § 4.2 we define the volume of a representation and use the existence of the bounded Euler class in even dimensions to prove in Theorem 4.4 a congruence relation for the volume of a representation.

Bounded volume and Euler classes
the vertical maps are isomorphisms .
Proof The fact that c R is an isomorphism follows from [3, Proposition 2.1] and the identifications between the (bounded) Borel and the (bounded) continuous cohomology.
To show that c Z is an isomorphism, we will do diagram chases in (2.10).

Surjectivity of c
. By exactness of the lines in (2.10), the image of α R in H n B (G, R /Z ) vanishes. And thus the same holds for the image of by restriction, and the latter vanishes, taking into account (2.2), since there are no SO(n)-invariant (n−1)-forms on R n and hence no G • -invariant differential (n − 1)-forms on H n .
As a consequence of the fact that c R is an isomorphism and that the volume of geodesic simplices in hyperbolic n-space is bounded, the volume cocycle defines also a bounded Borel class corresponding to the volume class ω n .
As an immediate consequence of Lemma 4.1 and the correspondence (3.5) we obtain: that has the following properties: (1) it is unique, and (2) under the change of coefficients Z → R it corresponds to

Remark 4.3 (1) With a slight abuse of notation we denote equally by
respectively to O(2m) and to SO(2m).
(2) If m = 1 and with the usual slight abuse of notation, the restriction is the usual bounded Euler class, where Isom(H 2 ) • is considered as a group of orientation preserving homeomorphisms of the circle.

Definition of volume and congruence relations
Let M be a complete finite volume hyperbolic n-dimensional manifold and let ρ : π 1 (M) → Isom + (H n ) be a homomorphism. Given a compact core N of M we consider ρ as a representation of π 1 (N ) and use the pullback via ρ in bounded cohomology together with the isomorphism to obtain a bounded singular class in H n b (N , R), denoted ρ * (ω b n ) by abuse of notation. Using the isometric isomorphism [2, Theorem 1.2], the volume of ρ is defined by If n = 2m, by the same abuse of notation, and by considering again the pullback in bounded cohomology via ρ, this time together with the isomorphism , which is a bounded singular integral class. Thus, denoting by δ b the connecting homomorphism in the long exact sequence in bounded singular cohomology (which is in fact an isomorphism), we have: Let M be a complete hyperbolic manifold of finite volume and even dimension n = 2m and N a compact core of M. If ρ : Proof In fact, (4.1) and (4.2) are part of the following diagram where the rows are obtained from the long exact sequence with X = ∂ N and X = N , induced by the change of coefficients in (2.8) and from the fact that H is amenable; the columns on the other hand follow from the long exact sequence in relative bounded cohomology associated to the inclusion of pairs (N , Let z be a Z-valued singular bounded cocycle representing ρ * (ε b 2m ) ∈ H 2m b (N , Z). Restricting z to the boundary ∂ N we obtain a Z-valued singular bounded cocycle z| ∂ N , which we know is a coboundary when considered as a R-valued cocycle since H 2m b (∂ N , R) = 0. Thus there must exist a bounded R-valued singular (2m − 1)cochain b on ∂ N such that where d is the differential operator on (bounded R-valued) singular cochains.
On the one hand, we note that since db is Z-valued, the cochain b mod Z is a R/Z-valued (2m − 1)-cocycle on ∂ N whose cohomology class is mapped to by the connecting homomorphism δ b in (4.2). On the other hand, define a bounded (N , R) and, since z R −db vanishes on ∂ N , we have actually constructed a cocycle representing the relative bounded class

Vanishing of the bounded Euler class on tori and the proof of Theorem 1.1
The goal of this section is to prove Theorem 1.  (SO(2m, 1), Z ) defined in the previous section restricts to a class on SO(2m, 1) • with trivial Z-coefficients. If m = 1 and ρ : Z → SO(2, 1) • is a homomorphism, then ρ * (ε b 2 ) ∈ H 2 b (Z, Z) ∼ = R/Z is the negative of the rotation number of ρ(1), [13]. In contrast to this, in higher dimension we have the following: . Before proving the theorem we need some information about Abelian subgroups of SO(2m, 1) • . To fix the notation, recall that Then the maximal compact subgroup K < SO(2m, 1) is the image of O(2m) under the homomorphism we define Since SO(2m, 1) • preserves each connected component of the two-sheeted hyperboloid We can now outline the proof of Theorem 5.1. According to Lemma 5.2 below, there are up to conjugation two cases to consider for ρ : Z 2m−1 → SO(2m, 1) • : (1) ρ takes values in P: then, building on Lemma 5.4, Lemma 5.5 shows that ε b 2m | P ∈ H 2m B,b (P, Z) vanishes and this implies Theorem 5.1 in this case.   Under the identification the point x 0 corresponds to (0, 0), so that and if we show that there are no SO(2m − 1)-invariant (2m − 2)-multilinear forms on R 2m−1 we will have shown that 2m−1 (M\P) P is one-dimensional. The fact that * 2m−1 (R 2m−1 ) is one-dimensional and the pairing * 2m−2 (R 2m−1 ) × (R 2m−1 ) * → * 2m−1 (R 2m−1 ) We show now that the exterior derivative on 2m−1 (M\P) P does not vanish. With the identification (5.1), the right translation R (t,x) by an element m(U )a(t)n(x) is given by In fact

Lemma 5.5 If m ≥ 2 then the restriction
vanishes.
Proof Considering the long exact sequence in bounded and in ordinary cohomology associated to (2.8), and taking into account Lemma 5.4 and the vanishing of bounded real cohomology of P (2.7), we obtain the diagram where we used Lemma 5.4 in the bottom row and the vanishing of bounded cohomology with real coefficients in the top row. Thus H 2m B,b (P, Z) ∼ = H 2m B (P, Z). If we show that ε 2m | P = 0, this will imply that ε b 2m | P = 0. To see that ε 2m | P = 0, by naturality of Wigner's isomorphism (2.4) and the fact that M is maximal compact in P, the restriction to M induces an isomorphism H 2m B (P, Z) ∼ = H 2m B (M, Z). Since however the Euler class ε 2m of SO(2m) restricted to 1 0 0 U : U ∈ SO(2m − 1) vanishes, we conclude that ε 2m | P = 0.
We deduce then using Lemma 5.5 that Theorem 5.1 holds if the image of ρ is contained in P. We must therefore turn to the case in which ρ(Z 2m−1 ) lies in T 0 . Let then be the projection on the i-th factor of T and let where ε 2 and ε b 2 are respectively the Euler class and the bounded Euler class of O (2). Observe that if L is compact, the long exact sequence (2.10) gives Since the ordinary Euler class is a characteristic class and T is a product, we have hence it follows from (5.2) that Observe that the image of SO(2) m in T is its connected component T • .
Thus we are reduced to analyze homomorphisms As before, since SO(2) m is compact and hence amenable, the connecting homomorphism is an isomorphism. We fix the orientation preserving identification SO(2) −→ R/Z , and, for m = 1, we define the homogeneous 1-cocycle and [Rot] ∈ H 1 B (SO(2), R/Z) the corresponding class. Then δ b ([Rot]) = ε b 2 and from this we deduce that if . As a result, it follows from the commutativity of the square (2) Y · · · Y ε (m) )) . (5.6) In order to prove the vanishing of the left hand side, we are going to show that ρ * (ϑ Y ε (2) Y · · · Y ε (m) ) = 0. To this end we use that the pairing is an isomorphism, where [Z 2m−1 ] ∈ H 2m−1 (Z 2m−1 , Z) ∼ = Z denotes the fundamental class. We will need the following Proof We first check that ∂z = 0 so that z is indeed a cycle. Let (1) , e σ (1) + e σ (2) , . . . , e σ (1) + · · · + e σ (n) ] .
Let ω R n ∈ H n (Z n , R) be the Euclidean volume class. Note that the volume class evaluates to 1 on the fundamental class, as the n-torus generated by the canonical basis has volume 1. A cocycle representing ω R n is given by V n : (Z n ) n+1 → R sending v 0 , v 1 , . . . , v n to the signed volume of the affine simplex with vertices in the lattice Z n ⊂ R n , that is In order to show that we need to evaluate V n on each summand of z. By definition, (2) , . . . , e σ (1) + · · · + e σ (n) ) (1) , e σ (1) + e σ (2) , . . . , e σ (1) + · · · + e σ (n) ) = sign(σ ) 1 n! det(e 1 , e 1 + e 2 . . . , e 1 + · · · + e n ) where we have used for the second equality the fact that the determinant is alternating with respect to line permutations. Summing up, we obtain thus proving the lemma.

Lemma 5.8 Let m ≥ 2. Then
Proof We use as a representative of ε 2 ∈ H 2 (SO (2), Z) the multiple − 1 2 of the orientation cocycle (see Remark 3.1). This cocycle takes values in 1 2 Z but represents an integral class and in particular evaluates to an integer on a fundamental class. Hence a representative for κ := ϑ Y ε (2) (SO(2) m , R/Z) is given by the cocycle mapping g 0 , . . . , g 2m−1 ∈ SO(2) m to the product where Rot 1 denotes the pullback to SO(2) m via the first projection π 1 of the homogeneous 1-cocycle Rot defined in (5.5), while Or j denotes the pullback to SO(2) m via π j of the orientation cocycle We now evaluate the pullback ρ * (κ) on the cycle z of Lemma 5.7 and, writing f i = ρ(e i ) ∈ SO(2) m , we obtain . . , f σ (1) · · · · · f σ (n) ). (5.10) To prove that this expression vanishes first observe that holds for any f 1 , f 2 , f 3 in SO(2) (but not in SL(2, R)). Specializing to f 1 = g −1 , f 2 = Id, f 3 = h and using that Or is alternating gives for any g, h in SO (2). Finally, using the invariance of Or, we can multiply the variables on the left hand side of the latter equality by f g, and the variables on the right hand side by f h to obtain for any f , g, h in SO(2). Thus from (5.10) and (5.11) and with τ 2 = (23), we obtain the asserted vanishing as (1) ) Proof of Theorem 5.1 Let ρ : Z 2m−1 → SO(2m, 1) • be a homomorphism. By Lemma 5.2 either ρ(Z 2m−1 ) < P up to conjugation and then ρ * (ε b 2m ) = 0 by Lemma 5.5 or ρ * (Z 2m−1 ) < T 0 up to conjugacy. Then either ρ(Z 2m−1 ) ⊂ T • and the vanishing follows from Lemma 5.6 or ρ(Z 2m−1 ) < T • , in which case Lemma 5.8 and (5.6) imply that ρ * (ϑ Y ε (2) Y · · · Y ε (m) ) = 0 and hence ρ * (ε b 2m ) = 0 by (5.7).
with the usual abuse of notation that . If now all the C i 's are tori, the above congruence relation and Theorem 5.1 imply that (−1) m 2 Vol(S 2m ) Vol(ρ) ∈ Z. In the general case, let p i : C i → C i be a covering of degree B 2m−1 that is a torus. Then which vanishes by Theorem 5.1.

Examples of nontrivial and non-maximal representations
In this section we give examples of volumes of representations. More precisely: -In § 6.1 we set ourselves in dimension 3. Here we show in particular that the volume of a Dehn filling of a finite volume hyperbolic manifold coincides with the volume of the filling representation. In fact Proposition 6.1 deals with a more general case. -In § 6.2, by glueing appropriately copies of a hyperbolic manifold of arbitrary dimensions with totally geodesic boundary, we construct manifolds M k and representations of π 1 (M k ) whose volume is a rational multiple of Vol(M k ).

Dimension 3: representations given by Dehn filling
Let M be a complete finite volume hyperbolic 3-manifold, which, for simplicity, we assume has only one cusp. If N is a compact core of M, its boundary ∂ N is Euclidean with the induced metric and hence there is an isometry ϕ : ∂ N → T 2 to a two-dimensional torus for an appropriate flat metric on T 2 . We obtain then a decomposition of M as a connected sum where the identification is via ϕ. We are now going to fill in a solid two-torus to obtain a compact manifold. To this end, let τ ⊂ ∂ N be a simple closed geodesic and let us choose a diffeomorphism ϕ τ : ∂ N → S 1 × S 1 , in such a way that ϕ τ (τ ) = S 1 × { * }. Then M τ is the connected sum identified via ϕ τ . Denote by j τ : N → M τ the canonical inclusion and by p : M → N the canonical projection given by the cusp retraction T 2 × R >0 → T 2 . The composition between the fundamental groups = π 1 (M) and τ = π 1 (M τ ). By Gromov-Thurston's (2π)-Theorem [16], for all geodesic curves τ for which the induced length is greater than 2π in the induced Euclidean metric on ∂ N , the compact manifold M τ admits a hyperbolic structure. Proposition 1.2 is then an immediate consequence of Proposition 6.1.
To prove the proposition, recall that by definition, the volume of the representation ρ τ is equal to where all maps involved can be read in the diagram below. We will start by defining a map F : H 3 (M τ ) → H 3 (N , ∂ N ) that will turn the diagram below into a commutative diagram (Lemma 6.3) and which will induce a canonical isomorphism (Lemma 6.2). N , ∂ N ).
two copies of C 0 two by two so as to obtain a connected closed hyperbolic manifold M k . Observe that Vol(M k ) = 2k Vol(M).
For any even 0 < < k, there are degree one maps f : M k → M k− obtained by folding copies of M in M k along its boundary. These maps send the last + 1 copies of M inside M k to the last copy of M in M k− as illustrated in the following picture for = 2: The induced representation of π 1 (M k ) obtained by the induced map on fundamental groups composed with the lattice embedding of π 1 (M k− ) in Isom(H n ) has volume equal to the volume of M k− , that is (k − )/k times the volume of the maximal representation. by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

is continuous for pointwise convergence on with control of norms and uniform convergence on compact sets on G n+1 .
Proof Let D = s( \G). Choose > 0 and C ⊂ D compact with μ(D\C) < . Then Now we write C α(r (gg 0 ), . . . , r (gg n ))dμ(g) Before we continue with the proof of the proposition, we need to show that for every compact subset K ⊂ G the number F K of translates of the fundamental domain D that K intersects is finite: Proof Set F := ∪ η∈ ηK and observe that F ∩ D = s( p(F)) is relatively compact by our choice of Borel section. Since γ K ∩ D = γ K ∩ (F ∩ D) and K and F ∩ D are relatively compact, the lemma follows by the discreteness of .
Using the simple set theoretical inequality valid for any sets B, E, E we obtain for each summand the estimate Thus The continuity of the right regular action of G on L 1 (G) concludes the proof of the proposition.
Note that the cocycle stays continues after transferring from Isom + (H n ) to Isom(H n ).
Since there are no coboundaries in degree n for Isom(H n )-equivariant continuous bounded cochains on H n , this implies that we have a strict equality between (6.4) and Vol(ρ) Vol(M) · ω n (g 0 K , . . . , g n K ) .