A Matsumoto-Mostow result for Zimmer's cocycles of hyperbolic lattices

As for the theory of maximal representations, we introduce the volume of a Zimmer's cocycle $\Gamma \times X \rightarrow \mbox{PO}^\circ(n, 1)$, where $\Gamma$ is a torsion-free (non-)uniform lattice in $\mbox{PO}^\circ(n, 1)$, with $n \geq 3$, and $X$ is a suitable standard Borel probability $\Gamma$-space. Our numerical invariant extends the volume of representations for (non-)uniform lattices to measurable cocycles and in the uniform setting it agrees with the generalized version of the Euler number of self-couplings. We prove that our volume of cocycles satisfies a Milnor-Wood type inequality in terms of the volume of the manifold $\Gamma \backslash \mathbb{H}^n$. This invariant can be interpreted as a suitable multiplicative constant between bounded cohomology classes. This allows us to characterize maximal cocycles for being cohomologous to the cocycle induced by the standard lattice embedding via a measurable map $X \rightarrow \mbox{PO}(n, 1)$ with essentially constant sign. As a by-product of our rigidity result for the volume of cocycles, we give a new proof of the mapping degree theorem. This allows us to provide a complete characterization of maps homotopic to local isometries between closed hyperbolic manifolds in terms of maximal cocycles. In dimension $n = 2$, we introduce the notion of Euler number of measurable cocycles associated to closed surface groups. It extends the classic Euler number of representations and it agrees with the generalized version of the Euler number of self-couplings up to a multiplicative constant. We show a Milnor-Wood type inequality whose upper bound is given by the modulus of the Euler characteristic. This gives an alternative proof of the same result for the generalized version of the Euler number of self-couplings. Finally, we characterize maximal cocycles as those which are cohomologous to the one induced by a hyperbolization.

1. Introduction 1.1. Historical background. The study of lattices in semisimple Lie groups of non-compact type has many applications both to algebra and geometry. One of the most remarkable properties is given by their rigidity. The investigation of rigidity of A crucial step in their proof is the description of the Euler number via a sort of proportionality principle [BFS13b,Lemma 4.10]. A careful reading of that proof and the involved diagrams shows that one can extend the notion of Euler number to arbitrary measurable cocycles Γ × X → PO • (n, 1), where Γ is a uniform lattice and (X, µ X ) is a standard Borel probability Γ-space. More precisely, one may drop both the restrictions on the probability space and on the target group to obtain results similar to the ones proved by Bader, Furman and Sauer [BFS13b] (see Section 5.2). We will refer to those results with our new hypothesis as generalized Bader-Furman-Sauer's results (see again Section 5.2).
In the case of torsion-free non-uniform lattices of PO(n, 1), one could study rigidity via maximal representations. This successful approach was initiated by Bucher, Burger and Iozzi [BBI13]. Their techniques are based on an accurate study of the bounded cohomology groups of PO(n, 1), still for n ≥ 3. The authors first show that the volume cocycle Vol n defined on S n−1 is bounded, alternating and PO(n, 1)equivariant. Then, following the functorial approach to bounded cohomology developed by Burger and Monod [Mon01,BM02], they show that Vol n canonically determines a cohomology class [Vol n ] which generates H n cb (PO(n, 1); R ε ) ∼ = R. Here R ε denotes the PO(n, 1)-module endowed with the sign action (see Section 3.2 for a precise definition). The crucial idea is that one may consider the pullback of the volume class along any representation ρ : Γ → PO • (n, 1) of a torsion-free non-uniform lattice Γ < PO • (n, 1). The pairing between the pullback class with the relative fundamental class of Γ\H n gives rise to a numerical invariant called volume of the representation ρ. The volume is invariant with respect to the conjugacy by elements of PO • (n, 1) and hence it provides a well-defined continuous function on the character variety X(Γ, PO • (n, 1)) with respect to the topology of pointwise convergence. Note that when n = 3 this numerical invariant agrees with the one introduced by both Dunfield [Dun99] and Francaviglia [Fra04] as the pullback of the volume form along any pseudo-developing map (a proof of the equivalence of these definitions was given by Kim [Kim16]).
Bucher, Burger and Iozzi [BBI13] also investigate maximal representations. They introduce a Milnor-Wood inequality showing that any representation ρ : Γ → PO • (n, 1) satisfies |Vol(ρ)| ≤ Vol(Γ\H n ). Hence the study of maximal representations can be translated in terms of rigidity properties of representations. Keeping the dimensional hypothesis n ≥ 3, they obtain that maximal representations must be conjugated to the standard lattice embedding Γ → PO • (n, 1) via an element in PO(n, 1). This is a suitable adaptation of Mostow-Prasad rigidity to the context of representations. In a more general setting, Francaviglia and Klaff [FK06] proved some similar rigidity results for their definition of volume of a representation Γ → PO(m, 1), this time assuming m ≥ n ≥ 3 (the rigidity of volume actually holds also at infinity, as proved by Francaviglia and the second author [FS18] for the real hyperbolic lattices: Moreover, the second author also showed that the rigidity holds for complex and quaternionic lattices [Sav18]). The interest in the study of volume of representations has recently grown, leading to the development of a rich literature [Poz15,KK16,Tho18,Fara,Farb].
On the other hand, one could naturally ask what happens for lattices in PSL(2, R) ∼ = Isom + (H 2 ). It is well-known that Mostow rigidity does not hold in the two-dimensional case (and, whence PSL(2, R) is not taut by itself). For instance, if Γ g is the fundamental group of a closed surface Σ g of genus g ≥ 2, we know via Teichmüller theory that there exists a space of real dimension 6g − 6 of inequivalent discrete and faithful representations of Γ g into PSL(2, R).
However, in dimension n = 2, working with uniform lattices, Bader, Furman and Sauer [BFS13b] show that PSL(2, R) is in fact taut relatively to the natural embedding of PSL(2, R) into Homeo + (S 1 ). Their strategy is similar to the proof of the 1-tautness of PO • (n, 1), whence via the study of the Euler number of maximal self-couplings.
By changing perspective and following the ideas of the study of representations, Ghys [Ghy87] noticed that the Euler class e ∈ H 2 (Homeo + (S 1 ); Z) is actually a bounded class and hence it determines a class e b ∈ H 2 b (Homeo + (S 1 ); Z) in the bounded cohomology group. Therefore, given a representation ρ : Γ g → Homeo + (S 1 ), one can still pullback the Euler class and define a numerical invariant via the Kronecker product. More precisely, Ghys defined the Euler number of ρ as the number eu(ρ) = ρ * b (e b ), [Σ g ] , where ·, · denotes the Kronecker product and [Σ g ] is the fundamental class of Σ g . Since Ghys showed that the pullback class is a total invariant for the semiconjugacy class of ρ, the Euler number is constant along the semiconjugacy class of ρ. Recall that ρ 1 , ρ 2 : Γ g → Homeo + (S 1 ) are semiconjugated if there exists an element f ∈ Homeo + (S 1 ) such that f (ρ 1 (γ)ξ) = ρ 2 (γ)f (ξ), for every γ ∈ Γ g . One can follow Ghys' approach to study maximal representations. Indeed, after the works of Milnor [Mil58] and Wood [Woo71], we know that any representation ρ satisfies the key estimate |eu(ρ)| ≤ |χ(Σ g )|. The challenging problem about the characterization of maximal representations was solved by Matsumoto in [Mat87]. Indeed, he proved that the maximal value of eu detects the semiconjugacy class of a hyperbolization π 0 : Γ g → PSL(2, R). We stress the analogy between this statement and the rigidity results [BBI13,BFS13b] when the dimension satisfies n ≥ 3. Note that this correspondence is not only formal, since one may in fact investigate these problems with similar techniques. Indeed, for instance, Iozzi [Ioz02] provided a new proof of Matsumoto's theorem using bounded cohomology.
1.2. Volume of measurable cocycles. In this paper, we propose to introduce a numerical invariant of measurable cocycles associated to torsion-free (non-)uniform lattices in PO • (n, 1). On the one hand, it can be considered as an extension of the generalized Bader-Furman-Sauer's Euler number to the non-uniform case (see Section 5.2). On the other hand, our invariant also extends the classical volume of representations to the more general setting of measurable cocycles (both in the uniform and non-uniform cases). Indeed given a representation ρ : Γ → PO • (n, 1) there exists a canonical way to define an associated measurable cocycle σ ρ : Γ × X → PO • (n, 1), for any standard Borel probability Γ-space (X, µ X ). We prove in Proposition 5.4, that the volume of ρ agrees with our invariant of σ ρ . For this reason, despite our invariant is inspired by Bader-Furman-Sauer's Euler number, we prefer to call it volume of measurable cocycles associated to a (non-)uniform lattice Γ < PO • (n, 1).
Let n ≥ 3 and let Γ be a torsion free (non-)uniform lattice in Γ < PO • (n, 1) and let σ : Γ × X → PO • (n, 1) be a measurable cocycle, where (X, µ X ) is a standard Borel probability Γ-space. We study measurable cocycles that admit an essentially unique equivariant boundary map φ : S n−1 ×X → S n−1 . We define the volume of the cocycle σ via the pullback of the volume cocycle Vol n along φ (see Definition 5.1). Much more generally we show in Section 4 that an equivariant boundary map allows to pullback an essentially bounded PO(n, 1)-equivariant Borel cocycle to obtain suitable classes in the bounded cohomology groups H • b (Γ; R). The first insight into the rigidity of measurable cocycles via our volume is described by Proposition 5.7, where we show that the volume is in fact invariant along the PO • (n, 1)-cohomology class. Moreover, following the general philosophy of maximal representations, we prove a Milnor-Wood type inequality in order to get a stronger rigidity result for measurable cocycles. Indeed, we extend not only both the Milnor-Wood type inequalities by Bucher, Burger and Iozzi [BBI13] and by Bader, Furman and Sauer [BFS13b], but we completely characterize maximal cocycles as follows: Theorem 1.1. Let Γ < PO • (n, 1) be a torsion-free non-uniform lattice. Let (X, µ X ) be a standard Borel probability Γ-space where µ X has no atoms. Let σ : Γ × X → PO • (n, 1) be a cocycle with an essentially unique boundary map φ : S n−1 ×X → S n−1 . Then, we have |Vol(σ)| ≤ Vol(Γ\H n ) , and equality holds if and only if σ is cohomologous to the cocycle associated to the standard lattice embedding i : Γ → PO • (n, 1) via a measurable function f : X → PO(n, 1) with essentially constant sign.
It would be nice to describe the family of cocycles having extremal values explicitly. To that end, we introduce the family of reducible cocycles and we prove that they have vanishing volume (see Example 5.12). Despite the theorem above is stated for non-uniform lattices, our construction still holds in the uniform case as explained in Remark 5.5 (compare with Section 5.2). Working with uniform lattices and using some results available in the literature [BBI13,BFS13b], we can describe some families of maximal cocycles: the one arising from maximal representations (and this in fact also holds in the non-uniform case), the one arising from ergodic integrable self-couplings (see Corollary 5.13) and, finally, the one arising from ergodic couplings of Isom(H n ), n ≥ 3 (see Corollary 5.13) .
We briefly explain here the idea of the proof of Theorem 1.1, which follows the scheme of a theorem by Bader, Furman and Sauer [BFS13b, Theorem B] (compare also with a result by the second author [Sav, Theorem 1]). A crucial step in the proof is given by the following proposition which expresses the volume of a measurable cocycle as a suitable multiplicative constant between bounded cohomology classes.
Proposition 1.2. Let n ≥ 3 and let G = Isom(H n ). Let Γ < G + be a torsionfree non-uniform lattice and let (X, µ X ) be a standard Borel probability Γ-space.
Using the formula reported above we show that when the cocycle σ is positively maximal, that is Vol(σ) = Vol(Γ\H n ), the associated measurable map φ x : S n−1 → S n−1 , φ x (ξ) := φ(ξ, x) sends almost every regular ideal tetrahedron to another regular ideal tetrahedron with the same orientation, for almost every x ∈ X. This is enough to prove that φ x coincides essentially with an isometry in PO • (n, 1) for almost every x ∈ X. Then, we use the previous result to construct a map f : X → PO • (n, 1) which is measurable by Fisher, Morris and Whyte [FMW04] and thus it realizes the desired conjugation of our cocycle. This strategy works fine in the case of positive maximal volume and can be suitably adapted to the case of negative maximal volume.
1.3. Maximal cocycles and local isometries. Thanks to Theorems 1.1 we have a complete description of maximal Zimmer's cocycles. In this paper we are going to show how the study of maximal cocycles can be suitably used to characterize maps between closed hyperbolic manifolds of dimension n ≥ 3 which are homotopic to local isometries. This characterization will be related to the well-known mapping degree theorem. A first proof of the mapping degree theorem was given by Kneser in the case of surfaces [Kne30]. He showed that given a map f : Σ g → Σ g ′ between two closed surfaces of genus g and g ′ , the following bound on the mapping degree of f holds Note that when Σ g and Σ g ′ are both endowed with a hyperbolic structure, by Gauss-Bonnet theorem we can substitute the Euler characteristics appearing in the estimate above with the areas of Σ g and Σ g ′ . More generally the mapping degree theorem states that the latter estimate holds in any dimension. Formally, given a map f : M 1 → M 2 between closed hyperbolic manifolds of the same dimension, we have the following bound Moreover, Thurston showed that in dimension n ≥ 3 the equality holds if and only if f is homotopic to a local isometry [Thu79,Theorem 6.4]. This generalizes Kneser's theorem [Kne30] to the higher dimensional case. There exist several different proofs of the mapping degree theorem. For instance Besson, Courtois and Gallot [BCG95,BCG96,BCG98] used the notion of natural map to get a proof in the case of locally symmetric rank one closed manifolds.
Their techniques were then extended by Connell and Farb to the higher rank case [CF03b,CF03a].
A different approach via ℓ 1 -homology and simplicial volume was introduced by Thurston [Thu79, Theorem 6.2.1 and Theorem 6.4] and Gromov [Gro82, Section 0.2]. The simplicial volume is a homotopy invariant of compact manifolds that measures the complexity of such a manifold in terms of the ℓ 1 -norm of (real) singular chains. It can be interpreted as the geometric dual of the theory of bounded cohomology. Their approach is based on the crucial proportionality principle [Thu79, Theorem 6,2] (see also [Mun80]), which says that the simplicial volume of a closed hyperbolic manifolds is proportional to the Riemannian one up to a multiplicative constant only depending on the dimension (see [Thu79,Gro82,BP92] for a detailed description on this topic).
Recently, many results about mapping degree via simplicial volume and bounded cohomology have arised in the literature [ However, it seems that their proof is still related to simplicial volume, once they have reproved Gromov and Thurston's proportionality principle. On the other hand, Derbez, Liu, Sun and Wang prove a stronger result [DLSW19, Proposition 3.1]. In a simplified version, given a map f : M 1 → M 2 between to closed hyperbolic manifold of the same dimension, they define the pullback of a representation ρ : π 1 (M 1 ) → PO • (n, 1). Then, the volume of the pullback representation turns out to be proportional to the one of ρ up to a multiplicative constant which agrees with the mapping degree of f .
In this paper, we prove the following technical result which will allow us to characterize local isometries in terms of maximal cocycles. Proposition 1.3. Let f : M 1 → M 2 be a continuous map with deg(f ) = 0 between closed hyperbolic n-manifolds, with n ≥ 3. Denote by Γ 1 and Γ 2 the fundamental groups of M 1 and M 2 , respectively. Consider a maximal cocycle σ : Γ 2 × X → PO • (n, 1). Then, we have where f * σ denotes the pullback cocycle of σ along f . This result can be interpreted as a reformulation of the result [DLSW19, Proposition 3.1] in the case of measurable cocycles. However, it turns out to encode useful information. First, it allows us to reprove the mapping theorem in dimension n ≥ 3, as shown in Corollary 6.2.
More surprisingly, Thurston's strict version of the mapping degree theorem [Thu79, Theorem 6.4] will allow us to describe maps homotopic to local isometries as the ones preserving maximal cocycles. More precisely, we are going to show that the pullback of a maximal cocycle is still maximal if and only if the map along which we are performing the pullback is homotopic to a local isometry. This is the content of the following: Proposition 1.4. Let f : M 1 → M 2 be a continuous map with deg(f ) = 0 between closed hyperbolic manifolds of the same dimension n ≥ 3. Let σ : Γ 2 ×X → PO • (n, 1) be a maximal cocycle. Then, f is homotopic to a local isometry if and only if f * σ is a maximal cocycle.
1.4. Euler number for measurable cocycles. In the same spirit of the volume invariant, in this paper we also provide the definition of Euler number of measurable cocycles defined in terms of uniform lattices in the spirit of the Euler number of representations. As mentioned in Remark 7.3 (compare with Section 5.2), this numerical invariant coincides with the generalized version of Bader, Furman and Sauer's Euler number [BFS13b] up to a multiplicative constant when we deal with (Γ, Γ)-couplings. One could also extend our Euler number to the case of non-compact surfaces, that is to non-uniform lattices. However, this situation seems to contain some subtleties and we prefer to postpone it to a forthcoming project.
Let π 0 : Γ g → PSL(2, R) be a hyperbolization and assume that Γ g acts on S 1 via π 0 . Consider a measurable cocycle σ : Γ g × X → Homeo + (S 1 ) with essentially unique boundary map φ : S 1 × X → S 1 . Here (X, µ X ) is again a standard Borel probability Γ g -space. As in the case of the volume, the existence of the map φ allows us to pullback the Euler cocycle and evaluate it on the fundamental class. This construction provides our Euler number eu(σ) associated to a measurable cocycle σ. If we restrict ourselves to cocycles associated to representations, we are able to prove in Proposition 7.4 that our invariant coincides with the classic Euler number of representations. Moreover, our Euler number is a well-defined numerical invariant since it is constant along the Homeo + (S 1 )-cohomology class of measurable cocycles as proved in Proposition 7.5. Our investigation on the relation between our Euler number of measurable cocycles and the Euler number of representations leads to a Milnor-Wood type inequality. Our approach to the study of our Euler number of measurable cocycles is substantially different from the one carried on by Bader, Furman and Sauer [BFS13b], but we are still able to provide a new proof of such an inequality (compare our Proposition 7.7 with the generalized version of Bader, Furman and Sauer's inequality [BFS13b, Corollary 4.9]). Finally, we are able to characterize maximal cocycles by proving an extension of Matsumoto's theorem in this setting: Theorem 1.5. Let Σ g be a closed surface of genus g ≥ 2 and let Γ g := π 1 (Σ g ). Let π 0 : Γ g → PSL(2, R) be a hyperbolization and assume that Γ g acts on S 1 via π 0 . Let (X, µ X ) be a standard Borel probability Γ g -space, where µ X has no atoms. Then for every cocycle σ : Γ g × X → Homeo + (S 1 ) with essentially unique boundary map φ : S 1 × X → S 1 , we have |eu(σ)| ≤ |χ(Σ g )| and equality holds if and only if σ is cohomologous to a cocycle induced by a hyperbolization.
We briefly mention that our proof follows the lines of a result by Bader, Furman and Sauer [BFS13b, Theorem C] (compare also with a result by the second author [Sav, Theorem 1]). In a similar way of what happens for the volume invariant, we are able to express the Euler number of a measurable cocycle as a suitable multiplicative constant between bounded cohomology classes.
Using the expression above, we show that when a cocycle σ is positively maximal, that is eu(σ) = |χ(Σ g )|, the associated map defined by φ x (ξ) := φ(ξ, x) is order preserving for almost every x ∈ X, where ξ ∈ S 1 . Hence, it essentially coincides with an element f ∈ Homeo + (S 1 ), for almost every x ∈ X. This construction provides the desired measurable function, which conjugates the cocycle σ with the one induced by the hyperbolization π 0 . The same technique works also for negatively maximal cocycles.
The main techniques that we develop in this paper could be extended to some other numerical invariants of representations. More precisely, in a forthcoming paper, we will study in a systematic way the theoretical setting in which one can extend numerical invariants of representations to measurable cocycles and we apply our results to the study of measurable cocycles associated to complex hyperbolic lattices.
Acknowledgements. We would like to thank Roman Sauer for the enlightening discussions about the topic of this paper and his interest on it. We also wish to thank Michelle Bucher, Marc Burger, Alessandra Iozzi and Stefano Francaviglia for useful comments and corrections on the paper. We finally thank Clara Löh for the helpful discussions about Borel measure spaces and her careful reading of the first draft of this paper.
Plan of the paper. In Section 2 we recall the general theory of Zimmer's cocycles. We describe the notion of couplings and the concept of boundary maps associated to measurable cocycles. In Section 3 we collect all the properties about bounded cohomology that we need in the sequel. In Section 3.1 we remind Burger-Monod's functorial approach to bounded cohomology. Then, Section 3.2 is devoted to the description of the continuous bounded cohomology of Isom(H n ). Here it appears the crucial definition of volume cocycle. In Section 3.3 it follows a brief exposition of the notion of Euler class and of the definition of the orientation cocycle. Since we will need later to work with relative bounded cohomology, Section 3.4 is dedicated to the study of its properties. Here it is described the fundamental pairing involving Kronecker product. Finally, in Section 3.5 the transfer maps are defined together with their properties.
Section 4 is mainly devoted to the proof of a fundamental technical lemma. Here, we describe how to perform the pullback of a cocycle in presence of measurable boundary map associated to a Zimmer's cocycle (see Definition 4.6).
In Section 5 we study the volume of measurable cocycles. Its definition appears in Section 5.1 (see Definition 3.9). Section 5.2 is mainly devoted to the comparison between our invariant in the uniform case and the generalized version of the Euler number defined by Bader, Furman and Sauer [BFS13b]. We investigate the properties of our volume in Section 5.3 and we prove that it extends the volume of representations (see Proposition 5.4). The volume is invariant on the Isom + (H n )-cohomology classes of measurable cocycle, as shown in Proposition 5.7. Section 5.4 is dedicated to the proof of our rigidity result, Theorem 1.1. Here we also define reducible cocycles and we show that they have vanishing volume (see Example 5.12). Two crucial results are the Milnor-Wood type inequality contained in Proposition 5.10 and the interpretation of the volume as a multiplicative constant given in Proposition 1.2. Finally, we conclude with Corollary 5.13 where we show that ergodic integrable self-couplings of uniform lattices and ergodic couplings of Isom(H n ), n ≥ 3, are maximal, whence they are conjugated to the cocycle associated to the standard lattice embedding.
In Section 6, we relate the volume of measurable cocycles with the degree of continuous maps between closed hyperbolic manifolds. After having introduced the notion of pullback cocycle with respect to a continuous map, we prove Proposition 1.3 which relates the volume of the pullback of maximal cocycle with the degree of the continuous map. This approach furnishes a different proof of the classic mapping degree theorem in Corollary 6.2. Finally, in Proposition 1.4 we characterize maps homotopic to local isometries via maximal cocycles.
Then we move to Section 7 where we study the Euler number of measurable cocycles, defined in Section 7.1 (see Definition 7.1). In Remark 7.3 we discuss the link between our Euler number and the generalized version of the one introduced by Bader, Furman and Sauer [BFS13b]. Section 7.2 is devoted to the study of the relation between our Euler number and the one associated to representations. More precisely in Proposition 7.4 we show that our invariant extends naturally the Euler number of representations. Following the line of the volume invariant, in Proposition 7.5 it is proved that the Euler number is constant along Homeo + (S 1 )cohomology classes of measurable cocycles. Finally, we conclude with Section 7.3 where we prove our main Theorem 1.5 using both the Milnor-Wood type inequality stated in Proposition 7.7 and the intepretation of the Euler number as multiplicative constant given in Proposition 1.6.

Zimmer's cocycles theory
In this section we are going to introduce some basic definitions and results about Zimmer's cocycle theory. For a detailed discussion of this subject we refer the reader to both Furstenberg [Fur81] and to Zimmer [Zim84].
Let G and H be two locally compact second countable groups. Let (X, µ) be a standard Borel measure space on which G acts in a measure-preserving way. In the sequel we will refer to it simply as standard Borel measure G-space. Denote by Meas(X, H) the space of measurable functions from X to H endowed with its natural topology.
At a first sight Equation (1) may appear quite mysterious, but it is just a suitable extension of the ordinary chain rule for derivates to this more general context. We mention here another equivalent approach for defining measurable cocycles. Since σ ∈ Meas(G, Meas(X, H)), Equation (1) is the characterization of Borel 1-cocycles with values in the G-module Meas(X, H) in the sense of Eilenberg-MacLane (see [FM77,Zim] for a similar description). Following this latter approach to measurable cocycles, we introduce the notion of cohomologous cocycles.
Definition 2.2. Let σ 1 , σ 2 : G × X → H be two measurable cocycles and let f : X → H be a measurable map. We denote by , for all g ∈ G and almost every x ∈ X. We say that σ 1 and σ 2 are cohomologous (or, equivalent) if σ 2 = σ f 1 , for some measurable map f : X → H. In order to make the reader more familiar with the notion of measurable cocycles, we are going now to discuss in details a couple of families of them. We mention that beyond our explicit examples, measurable cocycles are quite ubiquitous in mathematics. For instance, they naturally appear in differential geometry (as differentiation cocycles) and in measure theory (as Radon-Nykodim cocycles) We refer the reader to [Zim84, Examples 4.2.3 and 4.2.4] for an overview of such examples. However, along this paper we will be primarily interested in cocycles arising from either representation or self-couplings.
We begin by introducing cocycles associated to representations.
Definition 2.3. Let ρ : G → H be a continuous representation. Fix any standard Borel measure G-space (X, µ). The cocycle associated to the representation ρ , for every g ∈ G and almost every x ∈ X.
We warn the reader that the cocycle σ ρ just introduced depends actually both on ρ and on X. However, since the condition on X is not significant, we drop the X from the notation. Notice that when G is discrete any representation is automatically continuous and hence it admits an associated cocycle.
The previous definition provides a large family of measurable cocycles and it shows that the theory of representations sits inside the much wider framework of Zimmer's cocycles theory.
We now introduce another example of cocycles which arises from the study of self-couplings. We recall here the definition of coupling (compare with [BFS13b, Definition 1.1]): Definition 2.4. Let G and H be two locally compact second countable groups with Haar measures m G and m H , respectively. We will assume additionally that both G and H are unimodular. A coupling for G and H is the datum of a Lebesgue measure space (Ω, m Ω ) with a G × H-measurable, measure-preserving action such that there exist two finite measure spaces (X, µ) and (Y, ν) with measurable isomorphisms where ı is G-equivariant and  is H-equivariant, both with respect to the natural actions on the first factor. If there exists a coupling Ω for two groups G and H, we say that G and H are measure equivalent.
Given a coupling (Ω, m Ω ) of two groups G and H as above, one can construct two different cocycles associated to it. By the commutativity of the actions of G and H on Ω, we have a well-defined action of G on the space X. Indeed, X can be naturally identified with the space of H-orbits in Ω. Similarly, we obtain an action of H on the space Y . Both actions are measure preserving. For any g ∈ G and almost every h ∈ H and x ∈ X, there must exist h 1 ∈ H which depends on both g and x such that g(h, x) = (hh −1 1 , g.x) , where g.x denotes the action of g on X in order to distinguish it from the one on Ω.
Since h 1 only depends on both g and x, the previous formula leads to the definition of cocycles associated to a coupling.
Definition 2.5. Let (G, m G ) and (H, m H ) be two locally compact, unimodular, second countable groups with their respective Haar measures. Let (Ω, m Ω ) be a coupling for G and H. The right measure equivalence cocycle associated to the coupling is defined by for every g ∈ G and almost every x ∈ X and h ∈ H.
By interchanging the role of G and H, one can define the left measure equivalence cocycle for every h ∈ H and almost every g ∈ G and y ∈ Y . The left (respectively, right) measurable cocycle encodes the information about the action of After having introduced some examples of cocycles, we now move to the fundamental notion of boundary map in the sense of Furstenberg [Fur81]. Assume that both G and H admit a Furstenberg-Poisson boundary (see [Fur63] for a precise definition). For the convenience of the reader we mention here an example of Furstenberg-Poisson boundary that we will need in the sequel. Given a Lie group G of non-compact type, the Furstenberg-Poisson boundary can be naturally identified with the quotient G/P , where P is any minimal parabolic subgroup of G. More generally, any lattice Γ < G admits the previous quotient G/P as a natural Furstenberg-Poisson boundary. Denote by B(G) (respectively, by B(H)) the Furstenberg-Poisson boundary associated to G (respectively, H). Endow both boundaries with their natural Borel sigma algebras induced by the Haar sigma algebras on their respective groups.
Definition 2.6. Let σ : G × X → H be a measurable cocycle. We say that a measurable map φ : for every g ∈ G and almost every ξ ∈ B(G) and x ∈ X.
A boundary map for σ is a σ-equivariant measurable map φ.
Remark 2.7. In the definition above we assumed that both G and H admit a Furstenberg-Poisson boundary. However in the sequel we will need a slight more general notion of boundary map. Indeed we will work with the group Homeo + (S 1 ) which is not a Lie group when endowed with the discrete topology. To overcome this difficulty it will be sufficient to consider a suitable compact space on which the group acts. More precisely, consider G and H two locally compact secont countable groups. Assume that G admits a Furstenberg-Poisson boundary B(G) and that H acts continuously on a compact completely metrizable space Y . A generalized boundary map is a measurable map for every g ∈ G and almost every ξ ∈ B(G) and x ∈ X.
The existence and the uniqueness of a boundary map are both stricly related with the properties of the cocycle σ. For instance, any proximal cocycle admits an essentially unique boundary map. We refer the reader to [Fur81] for a discussion on that property.
Since boundary maps are defined in terms of measurable cocycles, it is natural to describe how they vary along a cohomology class.
Definition 2.8. Let σ : G × X → H be a measurable cocycle and assume it admits a boundary map φ : The boundary map associated to the twisted cocycle σ f is the measurable map defined by for almost every ξ ∈ B(G) and x ∈ X.
An easy computation shows that the map φ f is indeed σ f -equivariant with respect to the cocycle σ f . The same definition will hold also in the case of generalized boundary maps.

Bounded cohomology and first results
3.1. Burger-Monod's theory of continuous bounded cohomology. We recall here the functorial approach to continuous bounded cohomology introduced by Burger-Monod [BM02] (see also [Mon01]). We mention that some equivalent results in the special case of discrete groups date back to Ivanov [Iva87] (see also [Fri17]).
Usually, continuous bounded cohomology is introduced as the cohomology of the continuous bounded cochain complex (see Definition 3.6). However, it seems to be convenient here to anticipate its description in terms of strong resolutions by relatively injective modules. It is worth underling that the two approaches lead to the same definition, which specializes to the ordinary bounded cohomology in presence of discrete groups.
Let G be a locally compact second countable group. Let E be a Banach space and let π : G → Isom(E) be a representation. Then, a Banach G-module is a pair (E, π) endowed with an action of G on E via linear isometries, which is determined by π . We say that the Banach G-module is continuous if the map θ π : G × E → E defined by θ π (g, v) := π(g)v is so. Using the representation π, we define the maximal continuous submodule of E as follows where e ∈ G is the neutral element of G and · E denotes the Banach norm on E. By construction, the module CE is the largest submodule of E on which the action of G is continuous, whence the name maximal. A G-morphism ϕ : E → F is a linear G-equivariant map between Banach G-spaces.
Let i : E → F be a G-morphism that admits a left inverse ψ : F → E (not necessarily a G-morphism) with bounded norm ψ ≤ 1. Here · without subscript stands for the operator norm. The Banach G-module E is relatively injective if for every G-morphism α : A resolution of E is an exact augmented complex of Banach G-modules (E • , δ • ), where we set E 0 = E and E n = 0 for every n ≤ −1. Recall that a contracting homotopy for a resolution (E • , δ • ) of E is a collection of maps h n : CE n+1 → CE n such that h n ≤ 1 and h n • δ n + δ n−1 • h n−1 = id CE n for every n ∈ N. We say that a resolution (E • , δ • ) of E is strong if the maximal continuous subcomplex (CE • , δ • ) admits a contracting homotopy, where δ • denotes the restriction of the coboundary operator.
The complex of G-invariants associated to ( and δ • denotes the restriction of the coboundary operator. We are now ready to introduce the definition of continuous bounded cohomology (compare with [Mon01, Theorem 7.2.1]): Definition 3.1. Let G be a locally compact second countable group and let E be a Banach G-module. Given a strong resolution (E • , δ • ) of E by relatively injective G-modules, we define the continuous bounded cohomology of G with coefficients in Note that the previous definition suggests that we may compute the continuous bounded cohomology of G with coefficients in E via any strong resolution of E by relatively injective G-modules. For instance in Definition 3.6, we will compute the continuous bounded cohomology via the standard complex of continuous cochains.
We describe now a useful strong resolution of E by relatively injective modules that we will need in the sequel. To that end, let us assume that E is the dual of some Banach space. Then, it admits a natural weak- * topology and an associated Borel weak- * structure. For sake of simplicity we will assume that G admits a Furstenberg-Poisson boundary B(G) (for instance G is a Lie group or a lattice in a Lie group). We make this assumption since we do not want to introduce the notion of amenable G-action which is too technical and it will not be used later in the paper.
The following provides such useful strongly resolution of E by relatively injective modules: We endow it with the natural G-action given by is not yet a resolution of E. It must be completed just by adding the inclusion map E → L ∞ w * (B(G); E). Therefore, following [ We conclude this section by recalling that even strong resolutions which are not by relatively injective modules may encode useful information. For instance, let X be a measurable space on which G acts measurably.
and δ • denotes the standard homogeneous coboundary operator. We endow it with the structure of Banach G-module given by for every g ∈ G and every f ∈ B ∞ (X •+1 ; E).
We can complete the previous complex with the inclusion E → B ∞ (X; E). This provides a resolution of the Banach G-module of E. It is proved by Burger and Iozzi [BI02, Proposition 2.1] that the complex of bounded measurable functions (B ∞ (X •+1 ; E), δ • ) is in fact a strong resolution for the module of coefficients E. Unfortunately, since it is not a strong resolution by relatively injective modules, we cannot compute the continuous bounded cohomology of G with E-coefficients via (B ∞ (X •+1 ; E), δ • ). Nevertheless, by a result by Burger and Monod [BM02, Proposition 1.5.2], the cohomology of any strong resolution of the Banach G-module E maps naturally to the continuous bounded cohomology of G. This construction provides a canonical map This means that any bounded measurable cocycle f ∈ B ∞ (X •+1 ; E) G naturally determines a cohomology class in H • cb (G; E).

3.2.
Continuous bounded cohomology of G = Isom(H n ). Let n ≥ 3 and let G = Isom(H n ) be the group of isometries of the hyperbolic n-space. We are going to denote the same group also by PO(n, 1) (and PO • (n, 1) will be the connected component of the identity, that is the subgroup of orientation preserving isometries).
Our main reference about the continuous and the continuous bounded cohomologies of G is Bucher, Burger and Iozzi's paper [BBI13]. Let G + ≤ G denote the subgroup of orientation-preserving isometries. Since G + has index 2 as a subgroup of G, there exists a well-defined homomorphism Using the homomorphism ε, we endow R with the following structure of G-module: We will denote the G-module R either by R ε if it has the previous G-structure or by R if it is endowed with the trivial one.
We recall now the definition of continuous cohomology of G with R (ε) coefficients. The round brackets used as subscript for the real coefficients denote the fact the we may consider either the trivial action or the one twisted by the sign representation. We say that a continuous map f : for every g, g 0 , · · · , g • ∈ G. Let us consider the cochain complex and δ • is the classic coboundary operator defined by Similarly, we can consider the continuous bounded cochain complex and δ • is the restriction of the previous coboundary operator since it sends bounded cochains to bounded cochains.
Note that we may extend the continuous bounded cochain complex with the inclusion R (ε) → C cb (G; R ε ) G given by constant functions. Since the augmented exact complex (C cb (G •+1 ; R (ε) ) G ; δ • ) is a strong resolution of R (ε) by relative injective G-modules, it is immediate to check that the previous definition agrees with Definition 3.1.
A key feature of continuous bounded cohomology is that it is endowed with a natural L ∞ -seminorm. Indeed, the L ∞ -norm on continuous bounded cochains induces an L ∞ -seminorm in continuous bounded cohomology as follows ). The gap between the continuous and the continuous bounded cohomology groups may be studied via the comparison map. More precisely, the inclusion of cochain complexes , which we call comparison map. For the convenience of the reader, we will sometimes denote it by comp • G . Remark 3.7. As we mention in Section 3.1, if G is a discrete group all the theory agrees with the ordinary bounded cohomology of discrete groups (see for instance Frigerio's book [Fri17]). Hence, when we deal with that case, we will just refer to as the bounded cohomology of G and we will denote it simply by H • b (G; R (ε) ). Remark 3.8. It is immediate to check that the previous definition easily extends to any topological group acting on R either via the trivial action or via a homomorphism into the multiplicative group {−1, 1}. For instance, if we consider a representation ρ : Γ → G, then we endow R with the Γ-module structure given by Let us denote by R ρ the previous Γ-module structure on R. Then, since C • c(b) (ρ) is norm non-increasing, any representation induces pullback maps in continuous (bounded) cohomology: Since the boundary at infinity ∂H n of the hyperbolic n-space H n can be realized as quotient of either G or G + by a minimal parabolic subgroup, it may be identified with their Furstenberg-Poisson boundary. Therefore, applying [Mon01, Corollary 7.5.3] (see also Section 3.1), we know that H • cb (G; R (ε) ) can be computed via the following cochain complex where δ • denotes the homogeneous coboundary operator. For ease of notation along the paper we will identify ∂H n ∼ = S n−1 . We introduce now a cocycle which will be a key tool in our paper.
Definition 3.9. We define the volume cocycle Note that it can be seen as an element of both the spaces One of the peculiar features of the volume cocycle is that it is in fact the unique representative of its cohomology class. Indeed, following [BBI13, Lemma 1], the absence of (n − 1)-cocycles easily implies that cohomology groups H n cb (G; R ε ) and H n c (G; R ε ) coincide with the subspaces of cocycles in L ∞ ((S n−1 ) n+1 ; R (ε) ) G and C c ((H n ) n+1 ; R (ε) ) G , respectively. Here C c ((H n ) n+1 ; R (ε) ) G denotes the space of continuous real-valued G-equivariant functions on (n + 1)-tuples of points of H n and its cohomology is precisely H • c (G; R ε ) (see [Mon01]). The previous argument also shows that that the comparison map 3.3. The orientation cocycle and the Euler class. In this section we are going to introduce the notion of Euler class. We will use the orientation cocycle to get a preferred representative for this class. We refer the reader to [Ioz02] for a more detailed description about these notions.
Fix an orientation on S 1 and denote by Homeo + (S 1 ) the group of orientationpreserving homeomorphisms endowed with the discrete topology.
Consider Homeo Z (R) the group of the homeomorphisms of the real line commuting with the integer translation T : R → R given by T (x) = x + 1. This group can be thought of as the universal covering of Homeo + (S 1 ) and given an element f ∈ Homeo + (S 1 ) there exists a unique liftf ∈ Homeo Z (R) which satisfies f (0) ∈ [0, 1). The previous construction allows us to define a section as follows Associated to the section given above, one may define an integer valued 2-cocycle ε : (Homeo + (S 1 )) 2 → Z satisfying for every f, g ∈ Homeo + (S 1 ). The cocycle ε determines a bounded cohomology class e b ∈ H 2 b (Homeo + (S 1 ), R), which is independent of the chosen section s in the definition.
Definition 3.10. The previous cohomology class e b is called bounded Euler class.
Since we will need to represent explicitly e b via a measurable cocycle defined on (S 1 ) 3 , we introduce the orientation cocycle otherwise .
Notice that the function or is a bounded Homeo + (S 1 )-invariant Borel cocycle on S 1 and so it canonically determines a cohomology class, as proved by Buger and Iozzi [ Hence, we get the following definition.
Definition 3.11. We define the Euler cocycle as the representative of the Euler class given by 3.4. Relative cohomology. As we mentioned in the introduction our definition of volume of measurable cocycles will apply for both uniform and non-uniform torsionfree lattices Γ < G + . However, following [BBI13], when we deal with the nonuniform case, it is necessary to work with the relative bounded cohomology of spaces. For the convenience of the reader we recall here its definition. Let X be a topological space and let A ⊂ X be a subspace. Given a singular simplex σ : ∆ • → X, we denote by Im(σ) its image. Recall that the relative cohomology H • (X, A; R) is computed via the following cochain complex where C • (X; R) denotes the space of singular cochains in X and δ • is the usual coboundary operator. We define the bounded cochain complex as Definition 3.12. The relative bounded cohomology of the topological pair (X, A), denoted by H • b (X, A; R), is the cohomology of the bounded cocomplex (C • b (X, A; R), δ • ). Remark 3.13. The previous notion restricts to the bounded cohomology of a space X when we consider the pair (X, ∅).
The deep connection between bounded cohomology of spaces and bounded cohomology of groups is nicely express via Gromov's mapping theorem [Gro82] (see [FM,Corollary 4.4.5] for a topological proof by Frigerio and the first author following Gromov's approach and Ivanov's proof [Iva87, Theorem 4.1] using homological algebra). More precisely, let X be a topological space and let π 1 (X) denote its fundamental group. Then, there exists an isometric isomorphism in bounded cohomology Let n ≥ 3. The argument that follows actually holds also for n = 2, but we will not need this fact. Let now Γ < G + be a torsion-free non-uniform lattice and let M = Γ\H n . It is well known that M is a complete finite-volume hyperbolic manifold. Let N be a compact core of M , that is a compact subset of M such that M \ N consists of the disjoint union of finitely many horocyclic neighbourhoods of cusps. Let us call them E j with j = 1, · · · , k. Note that all compact cores of M are homotopy equivalent.
We are going to explain now how to evaluate an element α ∈ H n b (Γ; R) on the fundamental class [N, ∂N ] ∈ H n (N, ∂N ; R) of N via the Kronecker product. Recall that the Kronecker product is the following bilinear form To that end, we need to describe a map from H n b (Γ; R) to H n (N, ∂N ; R). First, let us consider the isometric isomorphism g M described in Gromov's mapping theorem. This allows us to map α to g M (α) ∈ H n b (M ; R). Hence, we are reduced to describe a map from H n b (M ; R) to H n (N, ∂N ; R). Recall that the short exact sequence Let n ≥ 2 and let G be as usual Isom(H n ). Let G + < G be the subgroup of orientation-preserving isometries. Consider Γ < G + a torsionfree non-uniform lattice. Let M = Γ\H n and let N be its compact core. In this section we are going to introduce two standard maps which allow to transfer the information contained in the bounded cohomology of Γ and in the cohomology of (N, ∂N ) to the continuous bounded and the continuous cohomologies of G, respectively. More precisely, following [BBI13] (see also [BIW10,BKK14]), we define two natural transfer maps . Remark 3.14. Note that we are going to define the transfer maps in the case of non-uniform lattices. However, the same techniques apply verbatim to the uniform case.
Let us begin by recalling the definition of trans Γ .
Definition 3.15. Let Γ and G as above. Let trans Γ be the following cochain map where ψ ∈ L ∞ ((S n−1 ) •+1 ; R) Γ , ξ 0 , · · · , ξ • ∈ S n−1 and µ is the normalized invariant probability measure on Γ\G. We define the transfer map trans Γ as the induced map in cohomology by trans Γ : It is easy to check both the G-equivariance of trans  Following [BBI13], let us denote by U = π −1 (M \ N ) the preimage under the universal covering projection π : H n → M of the finite disjoint union of horocyclic neighbourhoods of cusps. Since we can identify the relative differential forms Ω • (M, M \ N ; R) with the Γ-invariant relative differential forms Ω(H n , U ; R) Γ , we get the following definition: Definition 3.17. Let Γ and G be as above. We call trans dR the cochain map where ψ ∈ Ω • (H n , U ; R) Γ and µ is the normalized invariant probability measure on Γ\G.
We denote by H • ( trans • dR ) the induced map in coohomology: The same arguments for trans Γ shows that trans dR is indeed a cochain map and it is G-equivariant.
Since any G-invariant differential form on H n is closed by Cartan's Lemma, we may identify H • (Ω • (H n ; R ε ) G ) with Ω • (H n ; R ε ) G . Therefore, in order to define τ dR , it only remains to consider the Van Est isomorphism (see [Gui80, Corollary 7.2]) Definition 3.18. We define the transfer map as follows Remark 3.19. Note that τ dR is an isomorphism in top degree as mentioned by Bucher, Burger and Iozzi [BBI13]. fits in the following commutative diagram Remark 3.20. Note that when G = PSL(2, R), we will simply use R as coefficients in the cohomology groups appearing in Diagram (3). Indeed, in this special case the the trivial action on R agrees with the one twisted by the sign.

A technical lemma
The aim of this section is to prove a fundamental technical lemma that we need in the sequel. Let H be a locally compact second countable group and let Γ be a discrete group. We will suppose that Γ admits a Furstenberg-Poisson boundary B(Γ) and that H acts measurably on compact completely metrizable space Y .
Suppose that there exists a sign homomorphism ε : H → {−1, 1} and denote by H + = ker ε. For instance this is the case when either H = Isom(H n ) or H = Homeo + (S 1 ). In the latter case the sign homomorphism is trivial and H = H + .
Let (X, µ X ) be a standard Borel probability Γ-space with the additional assumptions on µ X to be atom-free. We will refer to this space in the sequel simply as a standard Borel probability Γ-space. Consider the measurable cocycle σ : Γ×X → H + and suppose that there exists an essentially unique boundary map φ : B(Γ)×X → Y . Here we are using the generalized version of boundary map introduced in Remark 2.7.
We will be interested in the sequel on the following problem: given a cocycle ψ ∈ B ∞ (Y •+1 ; R (ε) ) H we want to pullback it along φ, whence σ, in order to obtain a cocycle in L ∞ (B(Γ) •+1 ; R) Γ . Recall that R ε is endowed with the H-module structure induced by ε and R has trivial Γ-module structure. Our approach will follow some ideas that date back to Bader, Furman and Sauer's paper [BFS13b, Proposition 4.2] (these ideas are also discussed by the second author for the definition of the Borel invariant of measurable cocycles [Sav, Proposition 3.1]). However, we deal here with a more general version which will be suitable for our later computations. Recall that L ∞ (X) is a Γ-module endowed with the following action where in the last equality we consider R as a Γ-module endowed with the trivial action.
The fact that the previous map is a well-defined cochain map is the content of the following lemma.
Lemma 4.2. The pullback map C • (φ) along φ is a well-defined norm non-increasing cochain map.
Proof. It is immediate to check that C • (φ) is a norm non-increasing cochain map. Let us prove that it is well-defined, that is C • (φ)(ψ) is Γ-invariant. To that end, we identify L ∞ w * (B(Γ) •+1 ; L ∞ (X)) ∼ = L ∞ (B(Γ) •+1 × X) and we endow it with the diagonal Γ-action given by where γ ∈ Γ, ξ 0 , · · · , ξ • ∈ B(Γ) and x ∈ X, respectively. Since C • (φ)(ψ) lies in the Γ-module L ∞ (B(Γ) •+1 × X), we only have to prove that it is Γ-invariant with respect to the previous action. Given γ ∈ Γ, ξ 0 , · · · , ξ • ∈ B(Γ) and x ∈ X, we have where in the fourth step we used the σ-equivariance of φ and in the following one the H-equivariance of ψ. Moreover, the fact that σ takes value in H + allows us to conclude the chain of equalities. This proves that the pullback along φ is a well-defined, whence the thesis.
It is immediate to check that I • X is a cochain map. The following lemma shows that it is well-defined and norm non-increasing: Lemma 4.4. The integration cochain map I • X is well-defined and norm non-increasing. Proof. First we have to prove that for every ψ ∈ L ∞ w * (B(Γ) •+1 ; L ∞ (X)) Γ , the cocycle I • X (ψ) is Γ-invariant. Given γ ∈ Γ and ξ 0 , · · · , ξ • ∈ B(Γ), we have , where in the fourth step we used the fact that Γ acts in a measure-preserving way. This proves that the integration map is well-defined.
Let ψ ∈ L ∞ w * (B(Γ) •+1 ; L ∞ (X)) Γ and let us check that I • X is norm non-increasing. The following computation concludes the proof.
Remark 4.5. Note that the integration map is well-defined only on bounded cochains. There is no such a map for unbounded ones. This fact will be stressed later in the definition of volume of cocycles of uniform lattices (see Remark 5.3).
We are now able to construct our desired pullback.
Definition 4.6. Let Γ and H be two groups as at the beginning of this section. Fix a standard Borel probability Γ-space (X, µ X ). Consider a cocycle σ : Γ × X → H + with essentially unique boundary map φ : . We will need in the proof of the Milnor-Wood inequalities Propositions 5.10 and 7.7 that C • (Φ X ) is norm non-increasing. Lemmas 4.2 and 4.4 imply the following Lemma 4.7. The cochain map C • (Φ X ) is norm non-increasing.

Volume of Zimmer cocycles in
In this section we extend the classic notion of volume of representations to the much general setting of Zimmer's cocycles. Let Γ < G + = Isom + (H n ) be a torsionfree non-uniform lattice, where n ≥ 3 (this dimensional assumption will be assumed all along this section). Let M = Γ\H n be the complete finite-volume hyperbolic manifold with fundamental group Γ and let N be its compact core. Consider (X, µ X ) a standard Borel probability Γ-space. Let σ : Γ × X → G + be a measurable cocycle which admits an essentially unique boundary map φ : S n−1 × X → S n−1 , that is a σ-equivariant measurable map (see Definition 2.6).
Note that our setup only involves non-uniform lattices. However, all the definitions and results that we will discuss in this section still hold when we deal with uniform ones. For the convenience of the reader, we will stress the slight appropriate modifications needed to adapt our framework to uniform lattices. 5.1. Definition of volume. In this paragraph we define the notion of volume of cocycles. As proved in Section 4, there exists a natural way to pullback a cocycle c ∈ B ∞ ((S n−1 ) •+1 ; R ε ) G to L ∞ ((S n−1 ) •+1 ; R) Γ via the boundary map φ. Recall that the volume cocycle Vol n , given in Definition 3.9, is an element of B ∞ ((S n−1 ) n+1 ; R ε ) G . Therefore, we can pullback it from B ∞ ((S n−1 ) •+1 ; R ε ) G to L ∞ ((S n−1 ) •+1 ; R) Γ via the cochain map C n (Φ X ) as explained in Section 4. We define the volume class associated to σ as [C n (Φ X )(Vol n )] ∈ H n b (Γ; R) . Using the techniques introduced in Section 3.4, we can map the above element to H n b (N, ∂N ; R). Hence, the composition with the comparison map leads to our definition of volume of cocycles.
Definition 5.1. Let Γ < G + be a torsion-free non-uniform lattice and let (X, µ X ) be a standard Borel probability Γ-space. Let M = Γ\H n and let N be its compact core. Consider a cocycle σ : Γ × X → G + which admits an essentially unique boundary map φ. The volume of σ is defined as follows where ·, · is the Kronecker product, J n is the map introduced in Section 3.4 and [N, ∂N ] denotes the relative fundamental class of N . 5.2. The uniform case. As we mentioned in the introduction, our invariant is inspired by the Euler number of self-couplings introduced by Bader, Furman and Sauer [BFS13b]. Indeed, when we adapt the definition of our volume to uniform lattices and we restrict our attention to self-couplings, we recover Bader, Furman and Sauer's Euler number. Moreover, as explained at the end of this section, a careful reading of [BFS13b,Lemma 4.10] shows that one could define a generalized Euler number for arbitrary measurable cocycles, dropping both the assumptions on the target and on the measurable space. Then, this latter invariant agrees with our volume in the uniform case. We refer the reader to the end of this section for a discussion about this topic.
Let us explain now how the definition of volume in the non-uniform case can be suitably modified for dealing with uniform lattices. Since M = Γ\H n is now closed and aspherical, Poincaré Duality shows that the top dimensional cohomology group H n (M ; R) ∼ = H n (Γ; R) is isomorphic to R via the isomorphism obtained by the evaluation on the fundamental class [M ] of M . Therefore, we get the following definition: Definition 5.2. Let Γ < G + be a torsion-free uniform lattice and let (X, µ X ) be a standard Borel probability Γ-space. Let M = Γ\H n . Consider a cocycle σ : Γ × X → G + which admits an essentially unique boundary map φ. Then, we define the volume of σ to be where ·, · is the Kronecker product, g n M the isomorphism of Gromov's mapping theorem and [M ] denotes the fundamental class of M .
Remark 5.3. Note that also in the uniform case our definition of volume of cocycles defined in terms of uniform lattices involves the comparison map. At first glance, the presence of the comparison map may appear rather misleading with respect to the classic definition of volume of representations of uniform lattices (see [BBI13]). However, when we deal with measurable cocycles, we need to work with bounded cohomology. Indeed, any representation ρ : Γ → G + induces pullback maps in both continuous and continuous bounded cohomologies, which fit in the following commutative diagram: Unfortunately, as discussed in Remark 4.5 given a cocycle σ : Γ × X → G + we are only able to pullback the volume cocycle Vol n in bounded cohomology. Indeed, our construction is defined at the level of cochains via the following map . Since the integration cochain map I n X is well-defined only for bounded cocycles, our pullback provides a class in bounded cohomology.
As explained above (and in the introduction) one of the source of inspiration of our volume of measurable cocycles is the Euler number of self-couplings introduced by Bader, Furman and Sauer [BFS13b]. However, a careful reading of [BFS13b,Lemma 4.10] shows that one can easily extends Bader-Furman-Sauer's Euler number and their results to arbitrary measurable cocycles. In the sequel, we will refer to the extensions of both their invariant and their results, as generalized Euler number and generalized Bader-Furman-Sauer's results, respectively.
For the convenience of the reader, we explain here how to show that our volume of measurable cocycles agrees with the generalized Euler number in the case of uniform lattices. Before introducing the formal definition of the Euler number introduced by Bader, Furman and Sauer [BFS13b], it is convenient to recall the existence of an isometric isomorphism in bounded cohomology due to Monod and Shalom [MS06,Proposition 4.6]. Let Γ and Λ be two countable discrete groups. Recall by Definition 2.5 that given a (Γ, Λ)-coupling (Ω, m Ω ), there exists an associated right measure equivalence cocycle α Ω : Γ × Λ\Ω → Λ. It is proved by Monod and Shalom [MS06] that α Ω induces an isometric isomorphism which depends only on the coupling Ω, as suggested by the notation H • b (Ω). Bader, Furman and Sauer use the isometric isomorphism H • b (Ω) in order to define the Euler number associated to self-couplings [BFS13b]. More precisely, given a torsion-free uniform lattice Γ < G + and a (Γ, Γ)-coupling (Ω, m Ω ), the Euler number associated to Ω is defined as follows: Since Γ is a torsion-free uniform lattice in G + also the right measure equivalence cocycle α Ω : Γ × Γ\Ω → Γ takes values in G + . Therefore, we can compute its volume via Definition 5.2. We show now how our invariant in the uniform case agrees with the generalized version of Bader-Furman-Sauer's Euler number. For the convenience of the reader, we will denote by Γ ℓ and Γ r the left and the right copy of Γ, respectively.
Recall that in this setting, the existence of an essentially unique boundary map φ : S n−1 ×Γ ℓ \Ω → S n−1 for α Ω comes from the general theory of Furstenberg boundaries (see [Fur73,BM96,MS04]). Therefore, there is a well-defined notion of volume of the cocycle α Ω . Then, we can slightly modify the commutative diagram of [BFS13b,Lemma 4.10] in order to obtain the following one: Using the previous diagram, one can check that our volume is in fact the generalized Bader-Furman-Sauer's Euler number: . 5.3. Volume of cocycles vs. volume of representations. As we mentioned in the introduction, our aim is to define a notion of volume of measurable cocycles which extends the classic volume of representations introduced by Bucher, Burger and Iozzi [BBI13]. We formalize here this philosophical approach as follows: Proposition 5.4. Assume n ≥ 3. Let Γ < G + = Isom + (H n ) be a torsion-free nonuniform lattice. Let ρ : Γ → G + be a non-elementary representation with measurable boundary map ϕ : S n−1 → S n−1 . For any (X, µ X ) standard Borel probability Γ-space consider the measurable cocycle σ ρ : Γ × X → G + associated to ρ. Then, Proof. Recall that the measurable boundary map ϕ of ρ is essentially unique because of the doubly ergodic action of Γ on S n−1 . Therefore, we construct an essentially unique boundary map φ for σ ρ as follows: where ξ ∈ S n−1 and x ∈ X. Since the boundary map φ does not depend on the second variable x ∈ X, one can check that the following diagram commutes Recall that by Definition 4.6 we have C n (Φ X ) = I n X •C n (φ) and so the commutativity of the diagram above implies C n (Φ X )(Vol n ) = C n (ϕ)(Vol n ) .
Since Buger and Iozzi [BI02] proved that C n (ϕ)(Vol n ) is a natural representative of the cohomology class H n b (ρ)([Vol]), we get the following . This concludes the proof.
Remark 5.5. All the results regarding non-uniform lattices can be easily translated to the setting of uniform ones. For convenience of the reader we show here how to extend the previous result to that setting. Keeping the same notation of Proposition 5.4, we assume now that Γ < G + is a torsion-free uniform lattice. Note that the volume cocycle Vol n can be though of as an element of both C n cb (G; R ε ) and C n c (G; R ε ). In order to distinguish these two cases we will denote Vol b n and Vol n the continuous bounded and the continuous cocycles, respectively. Then, the proof in the presence of non-uniform lattices adapts to the uniform ones as follows: , where we used the commutativity of Diagram (4) completed with the dotted arrows.
Remark 5.6. We have just proved that the volume of a non-elementary representation ρ coincides with the volume of the associated cocycle σ ρ . On the other hand, it is well-known that elementary representations have vanishing volume. Therefore, it could be interesting to investigate which cocycles play the same role. As we will see later in Definition 5.11, reducible cocycles will provide an example of such a family.
Recall that the classic volume of representations satisfies the following property: for every g ∈ G and every representation ρ : Γ → G + . In particular, when g ∈ G + the volume of ρ is constant along the conjugacy class of the representation ρ.
Following the analogy between volume of cocycles and volume of representations, we prove now a similar property in our setting. Consider a measurable function f : X → G, where (X, µ X ) is a probability space. We define the sign of f as the function ε(f ) : X → {−1, 1} given by ε(f )(x) = ε(f (x)). In the case in which ε(f ) is almost everywhere constant, we will simply denote the real number identified with the essential image of ε(f ) by ε(f ) itself.
Then, we have the following: Proposition 5.7. Assume n ≥ 3. Let Γ < G + = Isom + (H n ) be a torsion-free nonuniform lattice and let (X, µ X ) be a standard Borel probability Γ-space. Consider a measurable function f : X → G such that its sign ε(f ) is almost everywhere constant. Let σ : Γ × X → G + be a measurable cocycle with an essentially unique boundary map φ : S n−1 × X → S n−1 . Then, we have In particular, if f : X → G + , we have that the volume is constant along the G +cohomology class of σ.
Proof. Recall by Definitions 2.2 and 2.8 that σ f is defined as for all γ ∈ Γ and for almost every x ∈ X, and its essentially unique boundary map associated is given by for almost every ξ ∈ S n−1 and x ∈ X. Let us denote by Φ f X the composition of the pullback along by φ f with the integration. Then, the volume of σ f can be computed as follows Note that in the step from the third and the fourth line, we used the hypothesis on the sign ε(f ) which is almost everywhere constant. Using the previous computation and the linearity of the Kronecker product we get the desired equality Using both Proposition 5.4 and Proposition 5.7 we obtain the following Corollary 5.8. Assume n ≥ 3. Let Γ < G + = Isom + (H n ) be a torsion-free nonuniform lattice. Fix (X, µ X ) a standard Borel probability Γ-space. If we denote by i : Γ → G + the standard lattice embedding, we have Vol(σ) = Vol(Γ\H n ) for every cocycle in the G + -cohomology class of σ i .
Proof. By [BBI13, Lemma 2] we know that the standard lattice embedding statisfies Vol(i) = Vol(Γ\H n ). Since the volume is constant along the G + -cohomology class by Proposition 5.7, the thesis follows.
Remark 5.9. At first sight it may seem rather unsatisfactory having the invariance of the volume only over a G + -cohomology class of the standard embedding lattice. However, if σ i denotes the cocycle associated to the standard lattice embedding, Proposition 5.7 ensures that |Vol(σ f i )| = Vol(Γ/H n ) , for every measurable function f : X → G with almost everywhere constant sign ε(f ). Therefore, it is immediate to check that, when we restrict to cocycles associated to representations, Proposition 5.7 and Corollary 5.8 provide a partial reformulation of Mostow rigidity from Bucher, Burger and Iozzi's point of view [BBI13]. This shows that Corollary 5.8 may be interpreted as an extension of classic ideas regarding Mostow rigidity of representations to the much wider setting of measurable cocycles. 5.4. Volume rigidity for Zimmer's cocycles. In this section we establish and study a Milnor-Wood type inequality for volume of cocycles. Remarkably, we will show that the maximal value is attained at cocycles which are cohomologous to the one associated to the standard lattice embedding i : Γ → G + via a measurable function with essentially constant sign.
Note that our result in the non-uniform setting extends the Milnor-Wood type inequality for volume of representations proved by Bucher, Burger and Iozzi [BBI13, Theorem 1]. Moreover, recall by Section 5.2 that in the uniform case our volume of measurable cocycles agrees with the generalized version of the Euler number introduced by Bader, Furman and Sauer [BFS13b]. Therefore, here we provide an alternative proof of the generalized version of the Milnor-Wood type inequality [BFS13b, Corollary 4.9] for measurable cocycles.
We denote by ω N,∂N the relative volume form in the relative de Rham complex, that is the unique form such that ω N,∂N , [N, ∂N ] = Vol(M ) . Then, by construction, we have τ n dR ([ω N,∂N ]) = ω n . Since Diagram (6) commutes, the following chain of equalities holds: N,∂N ]) . Since τ n dR is injective as recalled in Remark 3.19, we have Using the previous equality, it is immediate to check that the volume of σ may be expressed as follows As a consequence of Equation (7), taking the absolute value on both sides, we get Therefore, we reduce ourselves to prove that Recall that ω b n ∞ = Vol n ∞ because there are no coboundaries in H n cb (G; R ε ), as proved by Bucher, Burger and Iozzi [BBI13, Proposition 2]. Moreover, note that trans n Γ is norm non-increasing by definition. Therefore, the claim follows if we show that Note that, by the definition of · ∞ , we have Since we have already proved in Lemma 4.7 that C n (Φ X ) is norm non-increasing , we get whence the thesis.
Having introduced a Milnor-Wood type inequality, we are interested now in investigating maximal cocycles, that are cocycles of maximal volume. By Corollary 5.8 and Remark 5.9, we already know that all the cocycles which are cohomologous to the one associated to the standard lattice embedding via a measurable function with essentially constant sign are maximal. We spend the rest of this section in proving that in fact they are the only ones. This remarkable result provides our desired rigidity result for measurable cocycles.
To that end, we need to prove the formula reported in Proposition 1.2 which allows us to express the volume as a multiplicative constant. Our result will be a generalization of Bucher, Burger and Iozzi's formula [BBI13, Theorem 1.2]. Additionally, in the uniform case, our approach can be interpreted as an alternative proof of the generalized version Bader, Furman and Sauer's formula [BFS13b,Theorem 4.8] (see Section 5.2).
Proof of Proposition 1.2. As already proved in Proposition 5.10, we know that where ω b n = [Vol n ] ∈ H n cb (G; R ε ). A priori this equality holds at the level of cohomology classes but, as already explained in Section 3.2, we may identify H n cb (G; R ε ) with the space of bounded measurable cocycles on S n−1 . Hence, the above equality can be restated in terms of cocycles as follows: It is immediate to check that this provides precisely the desired formula.
Before going into the details of our rigidity Theorem 1.1, we define now a family of cocycles with vanishing volume.
Definition 5.11. Let n ≥ 3. Let Γ < G + = Isom + (H n ) be a torsion-free nonuniform lattice and let (X, µ X ) be a standard Borel probability Γ-space. Let σ : Γ × X → G + be a measurable cocycle. Let k < n, we say that the cocycle σ is reducible if it is cohomologous to a cocycle σ red : Γ × X → Isom + (H k ) through a measurable map f : X → Isom(H n ) with essentially constant sign. Here Isom + (H k ) is thought of as a subgroup of Isom + (H n ) via the upper left corner injection, that is where Id n−k is the identity matrix of order (n − k).
In the following example we show that reducible cocycles have volume equal to zero.
Example 5.12. Let σ be a reducible cocycle. By Proposition 5.7, without loss of generality we can work directly with a cocycle If we now assume that σ admits a measurable map φ : S n−1 × X → S n−1 , it should be clear that this map admits actually as target the (k − 1)-dimensional sphere S k−1 stabilized by Isom + (H k ). Hence for almost every x ∈ X we have a map φ x : S n−1 → S k−1 , φ x (ξ) := φ(ξ, x) whose image lies entirely in S k−1 . Moreover, since X is a standard Borel space, by [FMW04, Lemma 2.6] the map φ x is measurable for almost every x ∈ X.
We are now ready to discuss the proof of our main rigidity Theorem 1.1.
Proof of Theorem 1.1. Let σ i denote the cocycle associated to the standard lattice embedding. We have already proved in Corollary 5.8 and Remark 5.9 that cocycles cohomologous to σ i via a measurable function with essentially constant sign have maximal volume. It remains to prove the converse. Let σ : Γ × X → G + be a maximal measurable cocycle with associated essentially unique boundary map φ : S n−1 × X → S n−1 . Up to conjugacy by a suitable element g ∈ G, we may assume that the volume of σ is positive (see Proposition 5.7).
Fix a positive regular ideal tetrahedron with vertices ξ 0 , · · · , ξ n ∈ S n−1 and denote its volume by ν n .
Define the measurable map φ x : S n−1 → S n−1 by φ x (ξ) := φ(ξ, x). Notice that the hypothesis on X to be a standard Borel space implies that the map φ x is measurable for almost every x ∈ X, again by [FMW04, Lemma 2.6].
Following [BBI13, Proposition 5], we know that the equality stated in Proposition 1.2 actually holds for every ξ 0 , · · · , ξ n ∈ S n−1 . Hence, we get where µ is the normalized probability measure on the quotient Γ\G. Since the argument of the integral is bounded from above by ν n , the previous formula implies that ε(g) · Vol n (φ x (g −1 · ξ 0 ), · · · , φ x (g −1 · ξ n )) = ν n for almost every g ∈ Γ\G and almost every x ∈ X. Since φ is σ-equivariant, the previous formula holds in fact for almost every g ∈ G and for almost every x ∈ X. Following [BBI13, Proposition 6], this implies that φ x is almost everywhere equal to an orientation-preserving isometry f (x) ∈ G + . We now conclude the proof as describe by Bader, Furman and Sauer [BFS13b, Proposition 3.2]. For the convenience of the reader, we recall here the procedure for obtaining the desired conjugation. The previous construction allows us to define a map f : X → G + . Note that the mapφ : X → Meas(S n−1 , S n−1 ),φ(x) := φ x is measurable and its image lies entirely in the isometry group PO • (n, 1) ⊂ Meas(S n−1 , S n−1 ) (see also [BFS13b,Proposition 3.2]). Since by assumption X is a standard Borel space, the measurability of the map f follows now by [FMW04,Lemma 2.6]. Therefore, given γ ∈ Γ, on the one hand we have and on the other hand φ(i(γ)ξ, γx) = f (γx)(i(γ)ξ).
The previous computations show that whence σ is cohomologous to the cocycle associated to the standard lattice embedding i : Γ → G + .
We already know that among maximal cocycles we may find the ones associated to maximal representations. Since we already mentioned that all our results also hold in the uniform case, it is remarkable that in this situation we can describe other families of maximal cocycles. We anticipate here the family arising from ergodic integrable self-couplings and the ones coming from ergodic couplings of Isom(H n ). We postpone to Section 6 the discussion of the ones arising as pullback of maximal cocycles along maps homotopic to local isometries (see Proposition 1.4).
We recall first the definition of integrable self-coupling (see for instance [Sha00,BFS13a,BFS13b] for a discussion on this property).
Let Γ be torsion-free uniform lattice in G + and let ℓ : Γ → N be the length function associated to some word-metric on Γ. Keeping the notation of Section 5.3, consider a (Γ r , Γ ℓ )-coupling (Ω, m Ω ) with associated right measure equivalence co- where Γ ℓ \Ω is the normalized probability measure on Γ r \Ω.
Finally, given a (Γ, Λ)-coupling (Ω, m Ω ) with associated measure spaces (Γ\Ω, µ) and (Λ\Ω, ν) , we define the coupling index to be the following ratio: We are now ready to show that the volume of a cocycle associated to a coupling can be expressed in terms of the coupling index and the covolume of the target lattice. This result shows that both ergodic integrable self-couplings and ergodic couplings of Isom(H n ), with n ≥ 3, are maximal (compare with [BFS13b, Corollary 4.12]).
Proof. Denote by x Γ (respectively x Λ ) the cohomological fundamental class in H n (Γ; R) (respectively H n (Λ; R)). Let x b Γ ∈ H n b (Γ; R) be an element such that Let i Λ : Λ → G + be the standard lattice embedding associated to Λ. By construction (compare with [BFS13b,Corollary 4.12]) one can express the class x b Λ as follows .
If we now compute the volume of α Ω we get that In the computation above we used the fact the commutativity of Diagram (5) to pass from the first to the second line, Equation (8) to pass from the second to the third line and we applied [BFS13b,Theorem 4.11] (compare with [Thorem 5.12 and Corollary 1.11] [BFS13a]) to conclude. This shows the main equality.
Finally, as mentioned above, if (Ω, µ Ω ) = (Isom(H n ), m H ), with n ≥ 3, is an ergodic (Γ, Λ)-coupling, we know that the coupling is integrable (see [Sha00,Theorem 3.6] and [BFS13a, Remark 5.5]). Therefore, on the one hand, we have On the other hand, the coupling index is This concludes the proof.
Remark 5.14. In the uniform case, the previous corollary together with Theorem 1.1 show that the measurable cocycles associated either to ergodic integrable self-couplings or to couplings with respect to (Isom(H n ), m H ) are conjugated with the ones associated to maximal representations.

Maximal cocycles and mapping degree
Let M 1 and M 2 be two closed hyperbolic manifold of the same dimension n ≥ 3. We know by [Thu79,Theorem 6.4] (compare also with [BBI13, Corollary 1.3]), that given a continuous map f : and the equality holds if and only if f is homotopic to a local isometry.
We show here a result that characterizes maps homotopic to local isometries via the previous theorem using maximal cocycles (i.e. cocycles of maximal volume).
Let f : M 1 → M 2 be a continuous map. Denote by Γ 1 and Γ 2 the fundamental groups of M 1 and M 2 , respectively. Let π 1 (f ) : Γ 1 → Γ 2 be the induced map by f at the level of fundamental groups. Let us consider a measurable cocycle σ : Γ 2 × X → G + , where (X, µ X ) is a standard Borel probability Γ 2 -space. Note that (X, µ X ) can be also viewed as a standard Borel probability Γ 1 -space via the action induced by π 1 (f ). We define the pullback cocycle along f as for all γ ∈ Γ 1 and x ∈ X.
Lemma 6.1. The map f * σ is a measurable cocycle.
Given a continuous map f : M 1 → M 2 of non-vanishing degree, thanks to the work of [BM96,Fra09] there exists an essentially unique measurable map f : ∂H n → ∂H n which is π 1 (f )-equivariant. Now if we assume that σ admits a boundary map φ : S n−1 × X → S n−1 , the pullback cocycle along f admits the following boundary map: for all ξ ∈ S n−1 and x ∈ X.
Having introduced all the elements appearing in Proposition 1.3, we are now ready to prove it.
Proof of Proposition 1.3. Since σ is maximal, by the proof of Theorem 1.1 we know that the slices of the associated boundary map φ : S n−1 × X → S n−1 are isometries.
We have the following chain of equalities: , where we used [BI02] to implement the class H n b (π 1 (f ))[Vol n ] using the preferred representative C n ( f )(Vol n ) to pass from the third to the fourth line. Additionally we exploited the fact that comparison maps commute with the maps induced by representations to move from the fourth line to the fifth. This concludes the proof.
The previous proposition easily implies the mapping degree theorem, as shown in the following: Corollary 6.2. Let f : M 1 → M 2 be a continuous map with deg(f ) = 0 between closed hyperbolic manifolds of the same dimension n ≥ 3. Then, On the other, by Proposition 5.10 we have that whence the thesis.
Another remarkable application of Proposition 1.3 is the possibility to use the language of maximal cocycles in order to characterize continuous maps between closed hyperbolic manifolds that are homotopic to local isometries. This is the content of Proposition 1.4, whose proof is reported below.
Proof of Proposition 1.4. Suppose that f is homotopic to a local isometry. On the one hand, Thurston's strict version of the mapping degree theorem [Thu79, Theorem 6.4] implies that On the other hand, Proposition 1.3 implies This shows that f * σ is maximal. Suppose now the converse, that is f * σ is maximal. Then, Proposition 1.3 implies that The thesis now follows as a consequence of [Thu79, Theorem 6.4].

Euler number of Zimmer's cocycles of surface groups
Let Σ g a closed hyperbolic surface of genus g ≥ 2 and denote by Γ g := π 1 (Σ g ) its fundamental group. Fix a standard Borel probability Γ g -space (X, µ X ). In this section we want to define the Euler number associated to a measurable cocycle σ : Γ g × X → Homeo + (S 1 ) and study its rigidity property. This provides an extension of the study of maximal representations to the settings of measurable cocycles.
Since Γ g is a uniform lattice, we have already discussed in Section 5.2 how our approach leads to the generalized version of Bader-Furman-Sauer's Euler number. Here, we show in Remark 7.3 that our Euler number of measurable cocycles differs from the latter by a multiplicative constant. However, as mentioned above this section is mainly devoted to the extension of the Euler number of representations. Remarkably, the techniques that we develop here allows us to provide alternative proofs of the generalized version of some results by Bader, Furman and Sauer [BFS13b] for measurable cocycles.
Assume from now on that σ admits an essentially unique boundary map φ : S 1 × X → S 1 . It is important to underline that a priori there is no well-defined action of Γ g on the circle S 1 , hence we first need to fix a hyperbolization π 0 : Γ g → PSL(2, R) to specify this action. We will assume that Homeo + (S 1 ) has the discrete topology. Note that S 1 is a compact completely metrizable space on which Homeo + (S 1 ) acts in a measurable way. This means that for all along the section we consider generalized boundary maps as described in Remark 2.7. 7.1. Definition of Euler number. Recall by Section 3.3 that the Euler cocycle ǫ = −or/2 ∈ B ∞ ((S 1 ) 3 ; R) Homeo + (S 1 ) naturally determine a cohomology class in , and we call it the Euler class associated to σ. Let [Σ g ] ∈ H 2 (Σ g , R) be the fundamental class of Σ g . Similarly to what we have done in the previous section, we denote by comp 2 : H 2 b (Γ g ; R) → H 2 (Γ g ; R) the comparison map (which is non-trivial since Σ g is compact). We are now ready to define the Euler number associated to a cocycle.
Definition 7.1. Let Σ g be a closed surface of genus g ≥ 2 and let Γ g = π 1 (Σ g ). Let (X, µ X ) be a standard Borel probability Γ g -space. Fix a hyperbolization π 0 : Γ g → PSL(2, R) and assume that Γ g acts on S 1 via π 0 . Consider a cocycle σ : Γ g × X → Homeo + (S 1 ) with essentially unique boundary map φ : S 1 × X → S 1 . The Euler number eu(σ) associated to the cocycle σ is given by where ·, · is the Kronecker pairing and comp 2 , g 2 Σg denote the comparison map and the isomorphism appearing in Gromov's mapping theorem, respectively.
Remark 7.2. As discussed in Remark 5.3, recall that the comparison map also appears in our definition of volume of cocycles defined in terms of uniform lattices. For the same reason, we are forced to introduce the comparison map also when we define the Euler number of measurable cocycles.
Remark 7.3. We have already discussed in Section 5.2 how the results proved by Bader, Furman and Sauer [BFS13b] about (Γ, Γ)-couplings can be extended to their generalized versions (i.e. rephrased for arbitrarily measurable cocycles). Here, we show how our Euler number of measurable cocycle differs by a multiplicative constant from the generalized version of Bader-Furman-Sauer's Euler number. Via this remark, we will see in the sequel how some of our results provide alternative proofs of the generalized version of some results by Bader, Furman and Sauer [BFS13b].
Indeed, Bader, Furman and Sauer [BFS13b] consider the pullback of the Volume cocycle rather that of the Euler one. These cocycles are related by the following equation −2πǫ = Vol 2 .
Moreover, by standard result about Furstenberg boundaries theory, there exists a boundary map φ : S 1 × Γ g \Ω → S 1 associated to α Ω . As showed in Section 5.2, by a suitable modification of the commutative Diagram (5) to this context, we get 7.2. Euler number of cocycles vs. Euler number of representations. Recall by Definition 2.3 that, given any standard Borel probability Γ g -space (X, µ X ), every representation ρ : Γ g → Homeo + (S 1 ) induces naturally a cocycle σ ρ : Γ g × X → Homeo + (S 1 ).
We are going to prove, under an additional hypothesis on ρ, that the Euler number associated to σ ρ coincides with the classic Euler number associated to ρ. This shows that Definition 7.1 extends the classic notion of Euler number of representations to the wider theory of Zimmer's cocycles.
Proof. Thanks to the assumption about the existence of the essentially unique map ϕ, we can define a σ ρ -equivariant boundary map φ : S 1 × X → S 1 of σ ρ as follows: φ : S 1 × X → S 1 , φ(ξ, x) := ϕ(ξ) for almost every ξ ∈ S 1 and x ∈ X (see Section 2). Since the boundary map φ actually does not depend on the second variable x ∈ X, it is immediate to check that the two following pullback maps agree C 2 (Φ X )(ǫ) = C 2 (ϕ)(ǫ) .
It is a standard fact that the Euler number is constant along the semiconjugacy class of a representation ρ : Γ g → Homeo + (S 1 ) (see for instance [Ioz02]). We show now that a similar result still holds in the more general theory of Zimmer's cocycles. More precisely, the following proposition proves that the Euler number is constant on the cohomology class of a cocycle σ : Γ g × X → Homeo + (S 1 ).
Proof. We first note that we can consider the identity id S 1 : S 1 → S 1 as a measurable equivariant map with respect to the action of the hyperbolization π 0 on both the domain and the target space. Then, recall that if π 0 : Γ g → PSL(2, R) is a hyperbolization then eu(π 0 ) = χ(Σ g ), as shown by Iozzi [Ioz02]. The result now follows from Propositions 7.4 and 7.5. the orientation cocycle or is mapped to the standard volume form on H 2 normalized by π, that is ω/π. This means that the Euler class e = [−or/2] will be represented by −ω/2π. Now the volume form ω defines a natural volume form ω 0 on the surface Σ g endowed with the hyperbolic structure determined by the hyperbolization π 0 , that is π 0 (Γ g )\H 2 . Moreover the transfer map τ dR is injective in degree 2 and maps ω 0 in ω (see Remark 3.19).
By the commutativity of Diagram (11), we have and by injectivity of the transfer map we get If we now we evaluate both sides on the fundamental class [Σ g ] ∈ H 2 (Σ g ; R) we obtain where we used the Gauss-Bonnet theorem to pass from the third to the fourth line.
If we now take the absolute value on both sides, using Lemma 4.7, it follows that by an argument similar to the one exposed in Proposition 5.10. Note that ||e b || ∞ = ||ǫ|| ∞ because of the double ergodicity of the action of Γ g induced by π 0 . This concludes the proof.
We move forward to reach the proof of the generalized Matsumoto theorem for cocycles. In order to do this, we need first to prove Proposition 1.6 where we express the Euler number as a multiplicative constant between cocycles. The result we are going to prove will generalize [Ioz02, Proposition 1.7] (compare this result with the generalized version of Bader, Furman and Sauer's formula [BFS13b,Lemma 4.10]).
Proof of Proposition 1.6. We have already showed in the proof of Proposition 7.7 that and hence by linearity we argue that trans 2 [C 2 (Φ X )(or)] = eu(σ) χ(Σ g ) [or].
Since there are no essentially bounded PSL(2, R)-invariant cocycles on S 1 by the doubly ergodic action of Γ g induced by π 0 , the previous equality holds actually at the level of cocycles, that means trans 2 • C 2 (Φ X )(or) = eu(σ) χ(Σ g ) or, and the statement follows.
Thanks to Proposition 1.6, we are now ready to prove Theorem 1.5.