2.1 Amenability and measurable cocycles

In this section, we are going to recall some classic definitions related to both amenability and measurable cocycles. We start by fixing the following notation.

• • Let G be a locally compact second countable group endowed with its Haar measurable structure.

• • Let $(X,\mu )$ be a standard Borel measure G-space, that is a standard Borel measure space endowed with a measure-preserving G-action.

If $\mu $ is a probability measure, we will refer to $(X, \mu )$ as a standard Borel probability G-space. Given another measure space $(Y,\nu )$ , we denote by $\mathrm{Meas}(X,Y)$ the space of measurable functions from X to Y endowed with the topology of convergence in measure.

We recall now some definitions about amenability. We mainly refer the reader to the books by Zimmer [Reference ZimmerZim84, §4.3] and by Monod [Reference MonodMon01, §5.3] for further details about this topic.

Let $\operatorname {\mathrm{L}}^\infty (G;\operatorname {\mathrm {\mathbb {R}}})$ denote the space of essentially bounded real functions over G. Then, G acts on $\operatorname {\mathrm{L}}^\infty (G;\operatorname {\mathrm {\mathbb {R}}})$ as follows:

$$ \begin{align*}g.f (g_0) = f(g^{-1} g_0), \end{align*} $$

for all $g, g_0 \in \, G$ and $f \in \, \operatorname {\mathrm{L}}^\infty (G;\operatorname {\mathrm {\mathbb {R}}})$ .

Definition 2.2. A mean on $\operatorname {\mathrm{L}}^\infty (G;\operatorname {\mathrm {\mathbb {R}}})$ is a continuous linear functional

$$ \begin{align*}m \colon \operatorname{\mathrm{L}}^\infty(G;\operatorname{\mathrm{\mathbb{R}}}) \rightarrow \operatorname{\mathrm{\mathbb{R}}}, \end{align*} $$

such that $m(f)\geq 0$ whenever $f \geq 0$ and $m(\chi _G)=1$ , where $\chi _G$ denotes the characteristic function on G.

We say that a mean is left invariant if for all $g \in \, G$ and $f \in \, \operatorname {\mathrm{L}}^\infty (G;\operatorname {\mathrm {\mathbb {R}}})$ we have $m(g.f) = m(f)$ .

A group is amenable if it admits a left-invariant mean.

In what follows, we will need a more general notion of amenability which is related to group actions. In fact, amenable spaces and amenable actions will play a crucial role in the functorial approach to the computation of continuous bounded cohomology (§2.2). Following Monod’s convention, we begin by defining regular G-spaces [Reference MonodMon01, Definition 2.1.1].

Definition 2.4. Let G be a locally compact second countable group and let S be a standard Borel space with a measurable G-action which preserve a measure class. We say that $(S, \mu )$ is a regular G-space if the previous measure class contains a probability measure $\mu $ such that the isometric action $R \colon G \curvearrowright \operatorname {\mathrm{L}}^1(S, \mu )$ defined by

$$ \begin{align*}(R(g).f)(s) = f(g^{-1}.s) \frac{d g^{-1}\mu}{d\mu}(s) \end{align*} $$

is continuous. Here, $d g^{-1}\mu /d\mu $ denotes the Radon–Nikodým derivative.

The notion of regular G-spaces allows us to introduce the definitions of amenable actions and amenable spaces [Reference MonodMon01, Theorem 5.3.2].

Definition 2.6. Let G be a locally compact second countable group and let $(S,\mu )$ be a regular G-space. We say that the action of G on $(S,\mu )$ is amenable if there exists a continuous norm-one G-equivariant linear operator

$$ \begin{align*}p \colon\! \operatorname{\mathrm{L}}^\infty(G \times S;\operatorname{\mathrm{\mathbb{R}}}) \rightarrow \operatorname{\mathrm{L}}^\infty(S;\operatorname{\mathrm{\mathbb{R}}}), \end{align*} $$

with the following two properties: first, $p(\chi _{G \times S})=\chi _S$ ; second, for all $f \in \, \operatorname {\mathrm{L}}^\infty (G \times S)$ and for all measurable sets $A \subset S$ , we have $p(f \cdot \chi _{G \times A}) = p(f) \cdot \chi _A$ .

If the action by G on $(S, \mu )$ is amenable, then we say that $(S,\mu )$ is an amenable G-space.

We recall now the notion of measurable cocycles and some of their properties.

Definition 2.8. Let G and H be locally compact groups and let $(X,\mu )$ be a standard Borel probability G-space. A measurable cocycle (or, simply cocycle) is a measurable map $\sigma \colon G \times X \rightarrow H$ satisfying the following formula:

(1) $$ \begin{align} \sigma(g_1g_2,x)=\sigma(g_1,g_2.x)\sigma(g_2,x), \end{align} $$

for almost every $g_1,g_2 \in G$ and almost every $x \in X$ . Here, $g_2.x$ denotes the action by $g_2 \in G$ on $x \in \, X$ .

Associated with measurable cocycles, there exists the crucial notion of a boundary map.

Definition 2.9. Let G and H be two locally compact groups and let $Q \leq G$ be a closed amenable subgroup. Let $(X,\mu )$ be a standard Borel probability G-space and let $(Y,\nu )$ be a measure space on which H acts by preserving the measure class of $\nu $ . Given a measurable cocycle $\sigma \colon G \times X \rightarrow H$ , we say that a measurable map $\phi \colon G/Q \times X \rightarrow Y$ is $\sigma $ -equivariant if we have

$$ \begin{align*}\phi(g.\eta,g.x)=\sigma(g,x)\phi(\eta,x), \end{align*} $$

for almost every $g \in G,\eta \in G/Q$ and $x \in X$ .

A (generalized) boundary map associated with $\sigma $ is a $\sigma $ -equivariant measurable map.

We will make use of generalized boundary maps in §3.1, where we will explain how to compute the pullback in continuous bounded cohomology.

Remark 2.10. It is quite natural to ask when a (generalized) boundary map actually exists. Let $G(n)=\mathrm{Isom}(\operatorname {\mathrm {\mathbb {H}}}^n_{K})$ be the isometry group of the K-hyperbolic space, where K is either $\operatorname {\mathrm {\mathbb {R}}}$ or $\operatorname {\mathrm {\mathbb {C}}}$ . Given a lattice $\Gamma \leq G(n)$ , let us consider a standard Borel probability $\Gamma $ -space $(X,\mu _X)$ and a measurable cocycle $\sigma :\Gamma \times X \rightarrow G(m)$ . In the previous situation, Monod and Shalom [Reference Monod and ShalomMS04, Proposition 3.3] proved that if the cocycle $\sigma $ is non-elementary, then there exists an essentially unique boundary map,

$$ \begin{align*}\phi:\partial_\infty \operatorname{\mathrm{\mathbb{H}}}^n_{K} \times X \rightarrow \partial_\infty \operatorname{\mathrm{\mathbb{H}}}^m_{K}. \end{align*} $$

The notion of non-elementary cocycle relies on the definition of an algebraic hull (Definition 2.15) and this will be explained more carefully later in this paper.

Additionally, in the case of higher rank lattices, there are some relevant results about the existence of boundary maps. Indeed a key step in the proof of Zimmer’s super-rigidity theorem [Reference ZimmerZim80, Theorem 4.1] is to prove the existence of generalized boundary maps for Zariski dense measurable cocycles (see Definition 2.15).

Since equation (1) suggests that $\sigma $ can be interpreted as a Borel $1$ -cocycle in $\mathrm{Meas}(G,\mathrm{Meas}(X,H))$ [Reference Feldman and MooreFM77], it is natural to introduce the definition of cohomologous cocycles.

Definition 2.11. Let $\sigma \colon G \times X \rightarrow H$ be a measurable cocycle between locally compact groups. Let $f \colon X \rightarrow H$ be a measurable map. We define the twisted cocycle associated with $\sigma $ and f as

$$ \begin{align*}f.\sigma \colon G \times X \rightarrow H, \quad (f.\sigma)(g,x) := f(g.x)^{-1}\sigma(g,x)f(x), \end{align*} $$

for almost every $g \in G$ and almost every $x \in X$ .

We say that two measurable cocycles $\sigma _1,\sigma _2\colon G \times X \rightarrow H$ are cohomologous if there exists a measurable function $f \colon X \rightarrow H$ such that

$$ \begin{align*}\sigma_2=f.\sigma_1. \end{align*} $$

Similarly, we say that $\sigma _1$ and $\sigma _2$ are cohomologous modulo a closed subgroup $C \leq H$ if

$$ \begin{align*}\sigma_2=f.\sigma_1\! \mod C, \end{align*} $$

that is

$$ \begin{align*}\sigma_2(g,x) \cdot (f.\sigma_1(g,x))^{-1} \in C, \end{align*} $$

for almost every $g \in G, x \in X$ .

When a measurable cocycle $\sigma $ admits a generalized boundary map, then all its cohomologous cocycles share the same property.

Definition 2.12. Let $\sigma \colon G \times X \rightarrow H$ be a measurable cocycle with generalized boundary map $\phi \colon G/Q \times X \rightarrow Y$ . Given a measurable function $f \colon X \rightarrow H$ , the twisted boundary map associated with f and $\phi $ is defined as

$$ \begin{align*}f.\phi \colon G/Q \times X \rightarrow Y, \quad (f.\phi)(\eta,x) := f(x)^{-1}\phi(\eta,x), \end{align*} $$

for almost every $g \in G,\eta \in G/Q$ and $x \in X$ .

Representations provide special cases of measurable cocycles.

Definition 2.14. Let $\rho \colon G \rightarrow H$ be a continuous representation and let $(X,\mu )$ be a standard Borel probability G-space. The cocycle associated with the representation $\rho $ is defined as

$$ \begin{align*}\sigma_\rho \colon G \times X \rightarrow H, \quad \sigma_\rho(g,x)=\rho(g), \end{align*} $$

for every $g \in G$ and almost every $x \in X$ .

Given a representation $\rho \colon G \rightarrow H$ , one can obtain useful information about $\rho $ by studying the closure of the image $\overline {\rho (\Gamma )}$ as a subgroup of H. On the other hand, in general, the image of a measurable cocycle does not have any nice algebraic structure. Nevertheless, when H is assumed to be an algebraic group, we can give the following.

We will use the previous definition when we will work with totally real cocycles (§5) and when we will investigate the properties of cocycles with non-vanishing pullback (Theorem 6.1).

2.2 Bounded cohomology and its functorial approach

In this section, we are going to recall the definitions and the properties of both continuous and continuous bounded cohomology that we will need in what follows.

We first introduce continuous (bounded) cohomology via the homogeneous resolution and then, following the work by Burger and Monod [Reference Burger and MonodBM02, Reference MonodMon01], we describe it in terms of strong resolutions by relatively injective modules.

Definition 2.17. Let G be a locally compact group. A Banach G-module $(E, \pi )$ is a Banach space E endowed with a G-action induced by a representation $\pi \colon G \to \mathrm{Isom}(E)$ , or equivalently a G-action via linear isometries:

$$ \begin{align*} \theta_\pi \colon G \times E &\to E, \\ \theta_\pi (g, v) &:= \pi (g) v. \end{align*} $$

We say that a Banach G-module $(E, \pi )$ is continuous if the map $\theta (\cdot , v)$ is continuous for all $v \in \, E$ . Finally, we denote by $E^G$ the submodule of G-invariant vectors in E, that is, the space of vectors v such that $\theta (g, v) = v$ for all $g \in \, G$ .

Example 2.19. Let $(E, \pi )$ be a Banach G-module. Then, the space of continuous E-valued functions

$$ \begin{align*}\mathrm{C}_{c}^{\bullet}(G; E) := \{f \colon G^{\bullet+1} \rightarrow E \, | \, f\ \mbox{is continuous} \} \end{align*} $$

is a continuous Banach G-module with the following action:

(2) $$ \begin{align} g.f (h_1, \ldots, h_{\bullet +1}) = \pi(g) f(g^{-1} h_1, \ldots, g^{-1} h_{\bullet+1}), \end{align} $$

for all $g, h_1, \ldots , h_{\bullet+1} \in \, G$ .

The spaces of continuous E-valued functions give rise to a cochain complex $(\mathrm{C}_{c}^{\bullet}(G; E), \delta ^{\bullet})$ together with the standard homogeneous coboundary operator

$$ \begin{align*}\delta^{\bullet} \colon \mathrm{C}_{c}^{\bullet}(G; E) \rightarrow \mathrm{C}_{c}^{\bullet + 1}(G; E), \end{align*} $$

$$ \begin{align*}\delta^{\bullet} (f) (g_1, \ldots, g_{\bullet + 2}) := \sum_{j = 1}^{\bullet + 2} (-1)^{j - 1} f(g_1, \ldots, g_{j-1}, g_{j+1} \ldots, g_{\bullet + 2}). \end{align*} $$

Since this complex is exact, we are going to focus our attention on the subcomplex of G-invariant vectors $(\operatorname {\mathrm{C}}_c^{\bullet}(G; E)^G, \delta ^{\bullet})$ .

Since $(E, \lVert \cdot \rVert _E)$ is a Banach space, the Banach G-module $\operatorname {\mathrm{C}}_c^{\bullet}(G; E)$ has a natural $\operatorname {\mathrm{L}}^\infty $ -norm: For every $f \in \, \operatorname {\mathrm{C}}_c^{\bullet}(G; E)$ , we have

$$ \begin{align*}\lVert f \rVert_\infty := \sup \{ \lVert f(g_1, \ldots, g_{\bullet + 1} )\rVert_E \, | \, g_1, \ldots, g_{\bullet +1} \in \, G \}. \end{align*} $$

A continuous function is said to be bounded if its $\operatorname {\mathrm{L}}^\infty $ -norm is finite. Let $\operatorname {\mathrm{C}}_{cb}^{\bullet}(G; E) \subset \operatorname {\mathrm{C}}_c^{\bullet}(G; E)$ be the subspace of continuous bounded functions. By linearity, the coboundary operator $\delta ^{\bullet}$ preserves boundedness, so we can restrict $\delta ^{\bullet}$ to the space of continuous bounded G-invariant functions $\operatorname {\mathrm{C}}_{cb}^{\bullet}(G;E)^G$ . Then we get the following complex:

$$ \begin{align*}(\operatorname{\mathrm{C}}_{cb}^{\bullet}(G; E)^G, \delta^{\bullet}). \end{align*} $$

The $\operatorname {\mathrm{L}}^\infty $ -norm defined on cochains induces a canonical $\operatorname {\mathrm{L}}^\infty $ -seminorm in cohomology given by

$$ \begin{align*}\lVert f \rVert_\infty := \inf \{ \lVert \psi \rVert_\infty \ | \ [\psi]=f \}. \end{align*} $$

We say that an isomorphism between seminormed cohomology groups is isometric if the corresponding seminorms are preserved.

Beyond the difference determined by the quotient seminorm, one can study the gap between continuous cohomology and continuous bounded cohomology via the map induced in cohomology by the inclusion $ i \colon \! \operatorname {\mathrm{C}}_{cb}^{\bullet}(G; E)^G \rightarrow \operatorname {\mathrm{C}}^{\bullet}_{c}(G; E)^G$ . The resulting map

$$ \begin{align*}\operatorname{\mathrm{comp}}_G^{\bullet} \colon\! \operatorname{\mathrm{H}}_{cb}^{\bullet}(G; E) \rightarrow \operatorname{\mathrm{H}}_c^{\bullet}(G; E)\end{align*} $$

is called a comparison map.

In the following, we will need an alternative description of continuous bounded cohomology in terms of strong resolutions via relatively injective modules. Since we will not make an explicit use of these notions, we refer the reader to Monod’s book for a broad discussion on them [Reference MonodMon01, §§4.1 and 7.1]. The main result in this direction is the following.

We now describe a strong resolution via relatively injective modules which allows us to compute bounded cohomology isometrically. Let G be a locally compact second countable group. Let $(E, \pi , \lVert \cdot \rVert _E)$ be a Banach G-module such that E is the dual of some Banach space. This implies that E can be endowed with the weak- $^*$ topology and the associated weak- $^*$ Borel structure. Moreover, let $(S, \mu )$ be a regular G-space. We have the following.

Definition 2.25. We define the Banach G-module of bounded weak- $^*$ measurable E-valued functions on S to be the Banach space

$$ \begin{align*}&\mathcal{B}^{\infty}(S^{\bullet+1};E) := \{ f \colon S^{\bullet+1} \rightarrow E \, | \, f\ \text{is weak-}^*\ \mathrm{measurable}, \\&\qquad\qquad\qquad \mathop{\mathrm{sup}}\limits_{s_1,\ldots,s_{\bullet+1} \in S} \lVert f(s_1,\ldots,s_{\bullet+1}) \rVert_E < \infty \} \ \end{align*} $$

endowed with the following G-action $\tau $ :

$$ \begin{align*}(\tau(g) f) (s_1, \ldots, s_{\bullet+1}) := \pi(g) f(g^{-1}s_1, \ldots, g^{-1}s_{\bullet+1}) \end{align*} $$

for every $g \in \, G$ , $s_1, \ldots , s_{\bullet+1} \in \, S$ and $f \in \, \mathcal {B}^{\infty }(S^{\bullet+1};E)$ .

We define the Banach G-module of essentially bounded weak- $^*$ measurable E-valued functions on S to be

$$ \begin{align*}\mathrm{L}^\infty_{\mathrm{w}^*}(S^{\bullet+1};E) := \{ [f]_\sim \ | \ f \in \, \mathcal{B}^\infty(S^{\bullet+1};E)\}, \end{align*} $$

where $f \sim g$ if and only if they agree $\mu $ -almost everywhere and $[f]_\sim $ denotes the equivalence class of f with respect to $\sim $ .

Definition 2.27. Let us consider the situation above. We say that a(n essentially) bounded weak- $^*$ measurable function $f \colon S^{\bullet + 1} \rightarrow E$ is alternating if for every $s_1, \ldots , s_{\bullet+1} \in \, S$ we have

$$ \begin{align*}\operatorname{\mathrm{sign}}(\varepsilon) f(s_1, \ldots, s_{\bullet + 1}) = f(s_{\varepsilon(1)}, \ldots, s_{\varepsilon({\bullet + 1})}), \end{align*} $$

where $\varepsilon \in \mathfrak {S}_{\bullet+1}$ is a permutation whose sign is $\operatorname {\mathrm {sign}}(\varepsilon )$ .

Since the standard homogeneous operator $\delta ^{\bullet}$ preserves G-invariant (alternating) essentially bounded weak- $^*$ measurable functions up to a shift of the degree, we can consider the complex $(\mathrm{L}^\infty _{\mathrm{w}^*}(S^{\bullet+1};E), \delta ^{\bullet})$ . The following theorem shows when the previous complex computes isometrically the continuous bounded cohomology of G with E-coefficients.

As we have just discussed, one can compute continuous bounded cohomology by working with equivalence classes of bounded weak- $^*$ measurable functions. However, in some cases, it might be convenient to work directly with $\mathcal {B}^\infty (S^{\bullet + 1}; E)$ . Also in this case, the homogeneous coboundary operator sends (alternating) bounded weak- $^*$ measurable functions to themselves up to shifting the degree. Hence, we can still construct a complex $(\mathcal {B}^\infty (S^{\bullet + 1}; E), \delta ^{\bullet})$ . Unfortunately, the associated resolution of E is only strong in general [Reference Burger, Iozzi, Burger and MonodBI02, Proposition 2.1], so it cannot be used to compute the continuous bounded cohomology of G with E-coefficients. Nevertheless, one can obtain the following canonical map [Reference Burger, Iozzi, Burger and MonodBI02, Corollary 2.2]:

(3) $$ \begin{align} \mathfrak{c}^n \colon\! \operatorname{\mathrm{H}}^{n}(\mathcal{B}^\infty(S^{\bullet + 1}; E)^G) \rightarrow\operatorname{\mathrm{H}}^{n}(\operatorname{\mathrm{L}}_{\mathrm{w}^*}^\infty(S^{\bullet + 1}; E)^G) \cong \operatorname{\mathrm{H}}_{cb}^n(G; E), \end{align} $$

for every $n \in \, \mathbb {N}$ . This shows that each bounded weak- $^*$ measurable G-invariant function canonically determines a cohomology class in $\operatorname {\mathrm{H}}_{cb}^n(G; E)$ . The same result still holds in the situation of alternating functions.

In §3.1, we will tacitly use this result to show that the pullback of a bounded weak- ${}^*$ measurable G-invariant function lies in fact in $\operatorname {\mathrm{L}}^\infty _{\mathrm{w}^*}$ .

2.3 Transfer maps

In this section, we briefly recall the notion of transfer maps [Reference MonodMon01]. Let G be a locally compact second countable group and let $i \colon L \rightarrow G$ be the inclusion of a closed subgroup L into G. By functoriality, the inclusion induces a pullback in continuous bounded cohomology

$$ \begin{align*}\operatorname{\mathrm{H}}^{\bullet}_{cb}(i) \colon\! \operatorname{\mathrm{H}}_{cb}^{\bullet}(G; \mathbb{R}) \rightarrow \operatorname{\mathrm{H}}_{cb}^{\bullet}(L; \mathbb{R}). \end{align*} $$

A transfer map provides a cohomological left inverse to $\operatorname {\mathrm{H}}^{\bullet}_{cb}(i) $ . Assume that $L \backslash G$ admits a G-invariant probability measure $\mu $ (e.g. when L is a lattice of G), then we have the following.

Definition 2.32. We define the transfer cochain map as

$$ \begin{align*}\widehat{\mathrm{trans}}^{\bullet}_L \colon\! \operatorname{\mathrm{C}}_{cb}^{\bullet}(G;\operatorname{\mathrm{\mathbb{R}}})^L \rightarrow \operatorname{\mathrm{C}}_{cb}^{\bullet}(G;\operatorname{\mathrm{\mathbb{R}}})^G, \end{align*} $$

$$ \begin{align*}\widehat{\mathrm{trans}}^{\bullet}_L(\psi)(g_1, \ldots, g_{\bullet+1}) := \int_{L \backslash G} \psi(\overline{g}.g_1, \ldots, \overline{g}.g_{\bullet+1})\, d\mu(\overline{g}), \end{align*} $$

for every $(g_1, \ldots , g_{\bullet+1}) \in \, G^{\bullet+1}$ and $\psi \in \, \operatorname {\mathrm{C}}_{cb}^{\bullet}(G;\operatorname {\mathrm {\mathbb {R}}})^L$ . Here, $\overline {g}$ denotes the equivalence class of g in the quotient $L \backslash G$ .

The transfer map $\operatorname {\mathrm {trans}}_L^{\bullet}$ is the one induced in cohomology by $\widehat {\mathrm{trans}}^{\bullet}_L$ :

$$ \begin{align*}\operatorname{\mathrm{trans}}_L^{\bullet} \colon\! \operatorname{\mathrm{H}}_{cb}^{\bullet}(L;\operatorname{\mathrm{\mathbb{R}}}) \rightarrow \operatorname{\mathrm{H}}_{cb}^{\bullet}(G;\operatorname{\mathrm{\mathbb{R}}}). \end{align*} $$

We give now an alternative definition of the transfer map for essentially bounded weak- $^*$ measurable functions. Let Q and L be closed subgroups of a locally compact second countable group G. If Q is amenable, then the subcomplex of L-invariant essentially bounded functions on $G/Q$ computes the continuous bounded cohomology $\operatorname {\mathrm{H}}^{\bullet}_{cb}(L;\operatorname {\mathrm {\mathbb {R}}})$ (Remark 2.29 and Example 2.30). Hence, the new transfer map

$$ \begin{align*}\operatorname{\mathrm{trans}}_{G/Q}^{\bullet} \colon\! \operatorname{\mathrm{H}}^{\bullet}_{cb}(L; \mathbb{R}) \rightarrow \operatorname{\mathrm{H}}^{\bullet}_{cb}(G; \mathbb{R})\end{align*} $$

is the map induced in cohomology by the following:

$$ \begin{align*}\widehat{\mathrm{trans}}^{\bullet}_{G/Q} \colon\! \operatorname{\mathrm{L}}^\infty((G/ Q)^{\bullet+1};\operatorname{\mathrm{\mathbb{R}}})^L \rightarrow \operatorname{\mathrm{L}}^\infty((G/Q)^{\bullet+1};\operatorname{\mathrm{\mathbb{R}}})^G, \end{align*} $$

$$ \begin{align*}\widehat{\mathrm{trans}}^{\bullet}_{G/Q}(\psi)(\xi_1, \ldots, \xi_{\bullet+1}) := \int_{L \backslash G} \psi(\overline{g}.\xi_1, \ldots, \overline{g}.\xi_{\bullet+1})\, d \mu(\overline{g}), \end{align*} $$

for almost all $(\xi _1, \ldots , \xi _{\bullet+1}) \in \, (G/Q)^{\bullet + 1}$ and $\psi \in \, \operatorname {\mathrm{L}}^\infty ((G/Q)^{\bullet+1};\operatorname {\mathrm {\mathbb {R}}})^L$ .

The following commutative diagram completely describes the relation between the two transfer maps $\operatorname {\mathrm {trans}}^{\bullet}_L$ and $\operatorname {\mathrm {trans}}^{\bullet}_{G/ Q}$ [Reference Burger and IozziBI09, Lemma 2.43]:

(4)

Here, the vertical arrows are the canonical isomorphisms obtained by extending the identity $\mathbb {R} \rightarrow \mathbb {R}$ to the complex of continuous bounded and essentially bounded functions, respectively.

3.1 Pullback along measurable cocycles and generalized boundary maps

In this section, we introduce two different pullback maps in continuous bounded cohomology associated with a measurable cocycle. The first pullback will only depend on the measurable cocycle $\sigma $ . The second will be defined in terms of the generalized boundary map $\phi $ . We will show that under suitable conditions, the two definitions agree (Lemma 3.14). Despite a priori the first definition might appear more natural, we will mainly exploit the second pullback in the study of the rigidity properties of measurable cocycles.

Given a measurable cocycle $\sigma \colon L \times X \rightarrow G'$ , we define a pullback map from $\operatorname {\mathrm{C}}^{\bullet}_{cb}(G';\operatorname {\mathrm {\mathbb {R}}})^{G'}$ to $\operatorname {\mathrm{C}}^{\bullet}_b(L;\operatorname {\mathrm {\mathbb {R}}})^L$ as follows (compare with [Reference Moraschini and SaviniMS, Remark 14]).

Definition 3.2. In the situation of Setup 3.1, the pullback map induced by the measurable cocycle $\sigma $ is given by

$$ \begin{align*}\operatorname{\mathrm{C}}_b^{\bullet}(\sigma) \colon\! \operatorname{\mathrm{C}}^{\bullet}_{cb}(G';\operatorname{\mathrm{\mathbb{R}}}) \rightarrow \operatorname{\mathrm{C}}^{\bullet}_b(L;\operatorname{\mathrm{\mathbb{R}}}), \end{align*} $$

$$ \begin{align*}\psi \mapsto \operatorname{\mathrm{C}}^{\bullet}_b(\sigma)(\psi)(\gamma_1,\ldots,\gamma_{\bullet+1}) := \int_X(\psi(\sigma(\gamma_1^{-1},x)^{-1}),\ldots,\sigma(\gamma_{\bullet+1}^{-1},x)^{-1})\,d\mu_X(x). \end{align*} $$

Lemma 3.4. In the situation of Setup 3.1, the map $\operatorname {\mathrm{C}}^{\bullet}_b(\sigma )$ is a well-defined cochain map which restricts to the subcomplexes of invariant cochains. Hence, $\operatorname {\mathrm{C}}^{\bullet}_b(\sigma )$ induces a map in bounded cohomology

$$ \begin{align*}\operatorname{\mathrm{H}}^{\bullet}_b(\sigma):\operatorname{\mathrm{H}}^{\bullet}_{cb}(G';\operatorname{\mathrm{\mathbb{R}}}) \rightarrow \operatorname{\mathrm{H}}^{\bullet}_b(L;\operatorname{\mathrm{\mathbb{R}}}), \ \operatorname{\mathrm{H}}^{\bullet}_b(\sigma)([\psi]):=[ \operatorname{\mathrm{C}}^{\bullet}_b(\sigma)(\psi) ]. \end{align*} $$

Proof It is easy to check that $\operatorname {\mathrm{C}}^{\bullet}_b(\sigma )$ is a cochain map. Moreover, it sends bounded cochains to bounded cochains because $\mu _X$ is a probability measure.

It only remains to prove that $\operatorname {\mathrm{C}}^{\bullet}_b(\sigma )$ sends $G'$ -invariant continuous cochains to L-invariant ones. Let $\psi \in \operatorname {\mathrm{C}}^{\bullet}_{cb}(G';\operatorname {\mathrm {\mathbb {R}}})^{G'}$ and $\gamma ,\gamma _1,\ldots ,\gamma _{\bullet+1} \in L$ , then we have

$$ \begin{align*} &\gamma \cdot \operatorname{\mathrm{C}}^{\bullet}_b(\sigma)(\psi)(\gamma_1,\ldots,\gamma_{\bullet+1})\\[-2pt] &\quad\qquad=\operatorname{\mathrm{C}}^{\bullet}_b(\sigma)(\psi)(\gamma^{-1}\gamma_1,\ldots,\gamma^{-1}\gamma_{\bullet+1})\\[-2pt] &\quad\qquad=\int_X \psi(\sigma(\gamma^{-1}_1 \gamma,x)^{-1},\ldots,\sigma(\gamma^{-1}_{\bullet+1} \gamma,x)^{-1})\,d\mu_X(x)\\[-2pt] &\quad\qquad=\int_X \psi(\sigma(\gamma,x)^{-1}\sigma(\gamma^{-1}_1,\gamma.x)^{-1},\ldots,\sigma(\gamma,x)^{-1}\sigma(\gamma_{\bullet+1}^{-1},\gamma.x)^{-1})\,d\mu_X(x)\\[-2pt] &\quad\qquad=\int_X \psi(\sigma(\gamma,x)^{-1}\sigma(\gamma^{-1}_1,x)^{-1},\ldots,\sigma(\gamma,x)^{-1}\sigma(\gamma_{\bullet+1}^{-1},x)^{-1})\,d\mu_X(x)\\[-2pt] &\quad\qquad=\int_X\psi(\sigma(\gamma^{-1}_1,x)^{-1},\ldots,\sigma(\gamma_{\bullet+1}^{-1},x)^{-1})\,d\mu_X(x)\\[-2pt] &\quad\qquad=\operatorname{\mathrm{C}}^{\bullet}_b(\sigma)(\psi)(\gamma_1,\ldots,\gamma_{\bullet+1}), \end{align*} $$

where the second line is equal to the third one because of the definition of a measurable cocycle (Equation (1)). Then, the L-invariance of the measure $\mu _X$ shows that the third line is equal to the fourth one. Finally, the $G'$ -invariance of $\psi $ concludes the computation.

As anticipated, we now explain how to define a different pullback map via generalized boundary maps in the situation of Setup 3.1. This approach takes inspiration from a work by Bader, Furman, and Sauer [Reference Bader, Furman and SauerBFS13b, Proposition 4.2] and has already produced some applications in special settings (§3.5). We define the pullback along a generalized boundary map as the composition of two different maps defined in continuous bounded cohomology. The Banach space $\mathrm{L}^\infty (X) := \mathrm{L}^\infty (X; \mathbb {R})$ has a natural structure of Banach L-module given by the following L-action:

$$ \begin{align*}\gamma.f = f(\gamma^{-1}.x), \end{align*} $$

for all $\gamma \in L$ and $f \in \, \operatorname {\mathrm{L}}^\infty (X)$ . This leads to the following.

Definition 3.5. In the situation of Setup 3.1, the $\operatorname {\mathrm{L}}^\infty (X)$ -pullback along $\phi $ is the following map:

$$ \begin{align*}\operatorname{\mathrm{C}}^{\bullet}(\phi) \colon \mathcal{B}^\infty(Y^{\bullet + 1}; \mathbb{R})^{G'} \rightarrow \mathrm{L}_{\mathrm{w}^*}^\infty((G/Q)^{\bullet+1}; \operatorname{\mathrm{L}}^\infty(X))^L, \end{align*} $$

$$ \begin{align*}\operatorname{\mathrm{C}}^{\bullet}(\phi)(\psi) (\eta_1, \ldots, \eta_{\bullet+1}) := (x \mapsto \psi(\phi(\eta_1, x), \ldots, \phi(\eta_{\bullet+1}, x))), \end{align*} $$

where $\psi \in \, \mathcal {B}^\infty (Y^{\bullet + 1}; \mathbb {R})^{G'}$ , $\eta _1, \ldots , \eta _{\bullet+1} \in \, G/Q$ and $x \in \, X$ .

Proof Since $\operatorname {\mathrm{C}}^{\bullet}(\phi )$ is defined as a pullback, it is immediate to check that it is a norm non-increasing cochain map.

Let us show now that for every $\psi \in \, \mathcal {B}^\infty (Y^{\bullet + 1}; \mathbb {R})^{G'}$ , the cocycle $\operatorname {\mathrm{C}}^{\bullet}(\phi )(\psi )$ is L-invariant. First, by [Reference MonodMon01, Corollary 2.3.3], we can identify

$$ \begin{align*}\mathrm{L}_{\mathrm{w}^*}^\infty((G/Q)^{\bullet+1}; \operatorname{\mathrm{L}}^\infty(X))^L \cong \mathrm{L}^\infty((G/Q)^{\bullet+1} \times X)^L, \end{align*} $$

where the latter space is endowed with its natural diagonal L-action. Then, for almost every $x \in \, X$ , $\gamma \in \, L$ , and $\eta _1, \ldots , \eta _{\bullet+1} \in \, G/ Q$ , we have

$$ \begin{align*} \gamma \cdot \operatorname{\mathrm{C}}^{\bullet}(\phi)(\psi)(\eta_1, \ldots, \eta_{\bullet+1}) (x) &= \operatorname{\mathrm{C}}^{\bullet}(\phi)(\psi)(\gamma^{-1}.\eta_1,\ldots,\gamma^{-1}. \eta_{\bullet+1})(\gamma^{-1}.x)\\[-2pt] &=\psi(\phi(\gamma^{-1}.\eta_1, \gamma^{-1}.x), \ldots, \phi(\gamma^{-1}. \eta_{\bullet+1},\gamma^{-1}.x)) \\[-2pt] &= \psi(\sigma(\gamma^{-1}, x) \phi(\eta_1, x), \ldots, \sigma(\gamma^{-1}, x) \phi(\eta_{\bullet+1}, x))\\[-2pt] &= \psi(\phi(\eta_1, x), \ldots, \phi(\eta_{\bullet+1}, x)) \\[-2pt] &= \operatorname{\mathrm{C}}^{\bullet}(\phi)(\psi)(\eta_1, \ldots, \eta_{\bullet+1}) (x). \end{align*} $$

Here, we first used the definition of diagonal action, then the $\sigma $ -equivariance of $\phi $ , and finally the $G'$ -invariance of $\psi $ .

Since our final goal is to pullback a cocycle $\psi \in \, \mathcal {B}^\infty (Y^{\bullet + 1}; \mathbb {R})^{G'}$ along $\phi $ obtaining a new cocycle in $\mathrm{L}^\infty ((G/ Q)^{\bullet+1}; \mathbb {R})^L$ , we need to compose the $\operatorname {\mathrm{L}}^\infty (X)$ -pullback along $\phi $ with the integration map (compare with [Reference Bader, Furman and SauerBFS13b, Reference Moraschini and SaviniMS, Reference SaviniSava]).

Definition 3.7. In the situation of Setup 3.1, the integration map $\operatorname {\mathrm{I}}_X^{\bullet}$ is the following cochain map:

$$ \begin{align*}\operatorname{\mathrm{I}}_X^{\bullet} \colon \mathrm{L}_{\mathrm{w}^*}^\infty((G/Q)^{\bullet+1}; \operatorname{\mathrm{L}}^\infty(X))^L \rightarrow \mathrm{L}^\infty((G/Q)^{\bullet+1}; \mathbb{R})^L, \\ \end{align*} $$

$$ \begin{align*}\operatorname{\mathrm{I}}_X^{\bullet}(\psi)(\eta_1,\ldots,\eta_{\bullet+1}) := \int_X \psi(\eta_1, \ldots, \eta_{\bullet+1})(x)\, d\mu_X(x), \end{align*} $$

where $\psi \in \, \mathrm{L}^\infty ((G/Q)^{\bullet+1}; \operatorname {\mathrm{L}}^\infty (X))^L$ , $\eta _1, \ldots , \eta _{\bullet+1} \in \, G/Q$ , and $\mu _X$ is the probability measure on the standard Borel probability L-space X.

Proof Given a cocycle $\psi \in \, \mathrm{L}_{\mathrm{w}^*}^\infty ((G/Q)^{\bullet+1}; \operatorname {\mathrm{L}}^\infty (X))^L$ , it is easy to show that $\operatorname {\mathrm{I}}^{\bullet}_X(\psi )$ is L-invariant. Indeed, given $\eta _1,\ldots ,\eta _{\bullet+1} \in G/Q$ and $\gamma \in \, L$ , we have

$$ \begin{align*} \gamma. \operatorname{\mathrm{I}}_X^{\bullet}(\psi)(\eta_1, \ldots, \eta_{\bullet+1}) &= \int_X \psi(\gamma^{-1}.\eta_1, \ldots, \gamma^{-1}. \eta_{\bullet+1})(x)\, d\mu_X(x)\\[5pt] &= \int_X \psi(\eta_1, \ldots, \eta_{\bullet+1})(\gamma.x)\, d\mu_X(x) \\[5pt] &= \int_X \psi(\eta_1, \ldots, \eta_{\bullet+1})(x)\, d\mu_X(x) = \operatorname{\mathrm{I}}^{\bullet}_X(\psi)(\eta_1, \ldots, \eta_{\bullet+1}), \end{align*} $$

where we used the L-invariance of both $\psi $ and $\mu _X$ .

Since it is immediate to check that the integration map is also a norm non-increasing cochain map, we get the thesis.

We are now ready to define the pullback map along $\phi $ .

Definition 3.10. In the situation of Setup 3.1, the pullback map along the (generalized) boundary map $\phi $ is the following cochain map:

$$ \begin{align*}\operatorname{\mathrm{C}}^{\bullet}(\Phi^X) \colon \mathcal{B}^\infty(Y^{\bullet + 1}; \mathbb{R})^{G'} \rightarrow \mathrm{L}^\infty((G/Q)^{\bullet+1}; \mathbb{R})^L, \end{align*} $$

$$ \begin{align*}\operatorname{\mathrm{C}}^{\bullet}(\Phi^X) := \operatorname{\mathrm{I}}^{\bullet}_X \circ \operatorname{\mathrm{C}}^{\bullet}(\phi). \end{align*} $$

The fact that the pullback map induces a well-defined map in cohomology is proved in the following.

Proposition 3.12. In the situation of Setup 3.1, the pullback map $\operatorname {\mathrm{C}}^{\bullet}(\Phi ^X)$ is a norm non-increasing cochain map. Hence, it induces a well-defined map

$$ \begin{align*}\operatorname{\mathrm{H}}^{\bullet}(\Phi^X) \colon\! \operatorname{\mathrm{H}}^{\bullet}(\operatorname{\mathrm{\mathcal{B}}}^\infty(Y^{\bullet+1};\operatorname{\mathrm{\mathbb{R}}})^{G'}) \rightarrow \operatorname{\mathrm{H}}^{\bullet}_{b}(L;\operatorname{\mathrm{\mathbb{R}}}), \quad \operatorname{\mathrm{H}}^{\bullet}(\Phi^X)([\psi]):=[\operatorname{\mathrm{C}}^{\bullet}(\Phi^X)(\psi)]. \end{align*} $$

The same result still holds for the subcomplexes of alternating cochains.

Since we have introduced two different pullback maps in continuous bounded cohomology arising from measurable cocycles, it is natural to ask whether they agree. The following lemma completely describes the situation (compare with [Reference Burger, Iozzi, Burger and MonodBI02, Corollary 2.7]).

Lemma 3.14. In the situation of Setup 3.1, let $\psi \in \operatorname {\mathrm {\mathcal {B}}}^\infty (Y^{\bullet+1};\operatorname {\mathrm {\mathbb {R}}})^{G'}$ be a measurable cocycle. Then

$$ \begin{align*}\operatorname{\mathrm{C}}^{\bullet}(\Phi^X)(\psi) \in \operatorname{\mathrm{L}}^\infty((G/Q)^{\bullet+1};\operatorname{\mathrm{\mathbb{R}}})^L \end{align*} $$

is a representative of the class $\operatorname {\mathrm{H}}^{\bullet}_b(\sigma )([\psi ]) \in \operatorname {\mathrm{H}}^{\bullet}_{b}(L;\operatorname {\mathrm {\mathbb {R}}})$ .

Proof It is sufficient to consider the following commutative diagram [Reference Burger, Iozzi, Burger and MonodBI02, Proposition 1.2]:

where $\mathfrak{c}^{\bullet}$ is the map introduced in equation (3).

Finally, we show that the pullback along cohomologous measurable cocyles is the same (compare with [Reference Moraschini and SaviniMS, Propositions 13 and 20]).

Proposition 3.15. In the situation of Setup 3.1, let $f.\sigma \colon L \times X \rightarrow G'$ be a cocycle cohomologous to $\sigma $ with respect to a measurable map $f \colon X \rightarrow G'$ . Then, for every $\psi \in \mathcal {B}^\infty (Y^{\bullet + 1}; \mathbb {R})^{G'}$ , we have

$$ \begin{align*}\operatorname{\mathrm{C}}^{\bullet}(\Phi^X)(\psi) = \operatorname{\mathrm{C}}^{\bullet}(f.\Phi^X)(\psi).\end{align*} $$

Here, $\operatorname {\mathrm{C}}^{\bullet}(\Phi ^X)$ and $\operatorname {\mathrm{C}}^{\bullet}(f.\Phi ^X)$ denote the pullback maps along the associated boundary maps $\phi $ and $f.\phi $ , respectively.

Proof The boundary map $f.\phi $ associated with $f.\sigma $ is given by

$$ \begin{align*}f.\phi \colon G/Q \times X \rightarrow Y, \hspace{10pt} (f.\phi)(\eta, x) = f^{-1}(x) \phi(\eta, x), \end{align*} $$

for almost every $\eta \in \, G/Q$ and $x \in \, X$ (Remark 2.13). Hence, we have

$$ \begin{align*} \operatorname{\mathrm{C}}^{\bullet}(f.\Phi^X)(\psi)(\eta_1, \ldots, \eta_{\bullet+1}) &= \int_X \psi((f.\phi)(\eta_1, x), \ldots, (f.\phi)(\eta_{\bullet +1}, x))\, d\mu_X(x) \\ &= \int_X \psi(f^{-1}(x) \phi(\eta_1, x), \ldots, f^{-1}(x) \phi(\eta_{\bullet +1}, x))\, d\mu_X(x) \\ &= \int_X \psi( \phi(\eta_1, x), \ldots, \phi(\eta_{\bullet +1}, x))\, d\mu_X(x) \\ &=\operatorname{\mathrm{C}}^{\bullet}(\Phi^X)(\psi)(\eta_1, \ldots, \eta_{\bullet+1}), \end{align*} $$

for almost every $\eta _1, \ldots , \eta _{\bullet +1 } \in \, G/Q$ . This finishes the proof.

3.2 Pullback along generalized boundary maps vs. pullback of representations

In the situation of Setup 3.1, let $(X,\mu _X)$ be a standard Borel probability L-space and let $\rho \colon L \rightarrow G'$ be a representation. Then, there exists an associated measurable cocycle $\sigma _\rho \colon L \times X \rightarrow G'$ defined by $\sigma _\rho (\gamma , x) = \rho (\gamma )$ for every $\gamma \in \, L$ and $x \in \, X$ (Definition 2.14). If $\rho $ admits a $\rho $ -equivariant measurable map $\varphi \colon G/Q \rightarrow Y$ , the corresponding generalized boundary map of $\sigma _\rho $ is

$$ \begin{align*}\phi \colon G/Q \times X \rightarrow Y, \quad \phi(\eta, x) = \varphi(\eta), \end{align*} $$

for almost every $\eta \in \, G/Q$ and $x \in \, X$ .

As explained by Burger and Iozzi [Reference Burger, Iozzi, Burger and MonodBI02, Reference Burger and IozziBI09], one can implement the pullback map

$$ \begin{align*}\operatorname{\mathrm{H}}^{\bullet}_{cb}(\rho) \colon\! \operatorname{\mathrm{H}}^{\bullet}_{cb}(G'; \mathbb{R}) \rightarrow \operatorname{\mathrm{H}}^{\bullet}_{b}(L; \mathbb{R})\end{align*} $$

using a cochain map $\operatorname {\mathrm{C}}^{\bullet}(\varphi )$ defined by

$$ \begin{align*}\operatorname{\mathrm{C}}^{\bullet}(\varphi):\operatorname{\mathrm{\mathcal{B}}}^\infty(Y^{\bullet+1};\operatorname{\mathrm{\mathbb{R}}})^{G'} \rightarrow \operatorname{\mathrm{L}}^\infty((G/Q)^{\bullet+1};\operatorname{\mathrm{\mathbb{R}}})^L, \end{align*} $$

$$ \begin{align*}\psi \mapsto \operatorname{\mathrm{C}}^{\bullet}(\varphi)(\psi)(\eta_1,\ldots,\eta_{\bullet+1}):=\psi(\varphi(\eta_1),\ldots,\varphi(\eta_{\bullet+1})), \end{align*} $$

for almost every $\eta _1,\ldots ,\eta _{\bullet+1} \in G/Q$ .

The following result shows that the pullback associated with $\rho $ via $\varphi $ agrees with the one along $\phi $ . This property turns out to be fundamental to coherently extend the numerical invariants of representations to those of measurable cocycles (see [Reference SaviniSava, Proposition 3.4] and [Reference Moraschini and SaviniMS, Propositions 12 and 19]).

Proposition 3.17. In the situation of Setup 3.1, let $\rho \colon L \rightarrow G'$ be a representation which admits a $\rho $ -equivariant measurable map $\varphi \colon G/Q \rightarrow Y$ . Then, we have

$$ \begin{align*}\operatorname{\mathrm{C}}^{\bullet}(\Phi^X)=\operatorname{\mathrm{C}}^{\bullet}(\varphi). \end{align*} $$

Proof Since the boundary map $\phi $ associated with $\sigma _\rho $ does not depend on the second variable, it is immediate to check that the following diagram commutes:

from which the thesis.

3.3 Multiplicative formula

In this section, we prove how to deduce the multiplicative formula stated in Proposition 1.1. Some applications of the formula are then discussed in §3.5.

3.4 Multiplicative constants and maximal measurable cocycles

In this section, we are going to introduce the notion of a multiplicative constant. This definition will allow us to introduce maximal (measurable) cocycles and to investigate their rigidity properties.

In the situation of Setup 3.1, let $\psi ' \in \operatorname {\mathrm {\mathcal {B}}}^\infty (Y^{\bullet+1};\operatorname {\mathrm {\mathbb {R}}})^{G'}$ and let $\Psi =[\psi ] \in \operatorname {\mathrm{H}}^{\bullet}_{cb}(G;\operatorname {\mathrm {\mathbb {R}}})$ be represented by a bounded Borel cocycle $\psi \colon (G/Q)^{\bullet+1} \rightarrow \operatorname {\mathrm {\mathbb {R}}}$ . If $\operatorname {\mathrm{H}}^{\bullet}_{cb}(G;\operatorname {\mathrm {\mathbb {R}}})=\operatorname {\mathrm {\mathbb {R}}} \Psi $ , then Proposition 1.1 implies

(5) $$ \begin{align} &\int_{L \backslash G} \int_X \psi'(\phi(\overline{g}.\eta_1,x),\ldots,\phi(\overline{g}.\eta_{\bullet+1},x))\,d\mu_X(x)\,d\mu(\overline{g})\nonumber\\ &\quad=\lambda_{\psi',\psi}(\sigma)\psi(\eta_1,\ldots,\eta_{\bullet+1}) + \mathrm{cobound}. \end{align} $$

A particularly nice situation for the study of rigidity phenomena is when, in equation (5), there are no coboundary terms. For this reason, we are going to introduce the following notation.

Definition 3.22. We say that condition $(\operatorname {\mathrm {\mbox {NCT}}})$ (no coboundary terms) is satisfied when equation (5) reduces to

$$ \begin{align*} &\int_{L \backslash G} \int_X \psi'(\phi(\overline{g}.\eta_1,x),\ldots,\phi(\overline{g}.\eta_{\bullet+1},x))\,d\mu_X(x)\,d\mu(\overline{g})W\\ &\quad=\lambda_{\psi',\psi}(\sigma)\psi(\eta_1,\ldots,\eta_{\bullet+1}). \end{align*} $$

Remark 3.24. The condition $(\operatorname {\mathrm {\mbox {NCT}}})$ has the following equivalent reformulation via cochains:

$$ \begin{align*}\widehat{\mathrm{trans}}^{\bullet}_{G/Q} \circ \operatorname{\mathrm{C}}^{\bullet}(\Phi^X)(\psi')=\lambda_{\psi',\psi}(\sigma)\psi. \end{align*} $$

If condition $(\operatorname {\mathrm {\mbox {NCT}}})$ is satisfied, then there exists an explicit upper bound for the multiplicative constant $\lambda _{\psi ',\psi }(\sigma )$ .

Proposition 3.25. In the situation of Setup 3.1, let $\psi ' \in \operatorname {\mathrm {\mathcal {B}}}^\infty (Y^{\bullet+1};\operatorname {\mathrm {\mathbb {R}}})^{G'}$ and let $\Psi =[\psi ] \in \operatorname {\mathrm{H}}^{\bullet}_{cb}(G;\operatorname {\mathrm {\mathbb {R}}})$ be represented by a bounded Borel cocycle $\psi \colon (G/Q)^{\bullet+1} \rightarrow \operatorname {\mathrm {\mathbb {R}}}$ . If condition $(\mathrm{NCT})$ is satisfied, then we have

$$ \begin{align*}|\lambda_{\psi',\psi}(\sigma)| \leq \frac{\lVert \psi' \rVert_\infty}{\lVert \psi \rVert_\infty}. \end{align*} $$

Proof By Remark 3.24, we know that

$$ \begin{align*}\widehat{\mathrm{trans}}^{\bullet}_{G/Q} \circ \operatorname{\mathrm{C}}^{\bullet}(\Phi^X)(\psi')=\lambda_{\psi',\psi}(\sigma)\psi. \end{align*} $$

Since by Proposition 3.12 $\widehat {\mathrm{trans}}^{\bullet}_{G/Q}$ and $\operatorname {\mathrm{C}}^{\bullet}(\Phi ^X)$ are norm non-increasing maps, the left-hand side admits the following estimate:

$$ \begin{align*}\lVert \widehat{\mathrm{trans}}^{\bullet}_{G/Q} \circ \operatorname{\mathrm{C}}^{\bullet}(\Phi^X)(\psi') \rVert_\infty \leq \lVert \psi' \rVert_\infty . \end{align*} $$

Hence, we get

$$ \begin{align*}|\lambda_{\psi',\psi}(\sigma)| \lVert \psi \rVert_\infty \leq \lVert \psi' \rVert_\infty, \end{align*} $$

as desired.

Using the previous upper bound, we introduce the following.

Definition 3.26. In the situation of Setup 3.1, assume that condition $(\operatorname {\mathrm {\mbox {NCT}}})$ is satisfied. We say that a measurable cocycle $\sigma \colon L \times X \rightarrow G'$ is maximal if its multiplicative constant $\lambda _{\psi ',\psi }(\sigma )$ attains the maximum value:

$$ \begin{align*}\lambda_{\psi',\psi}(\sigma) = \frac{\lVert \psi' \rVert_\infty}{\lVert \psi \rVert_\infty}. \end{align*} $$

For every representation $\pi \colon G \to G'$ , we denote the restriction of $\pi $ to L as $\pi |_L \colon $ $L \to G'$ . We prove now that under suitable assumptions, maximal cocycles can be trivialized, that is, they are cohomologous to a suitable representation $\pi |_L \colon L \rightarrow G'$ .

Remark 3.29. More precisely, the theorem shows the existence of a measurable map $f \colon X \rightarrow G'$ , such that, for all $\gamma \in L$ and almost every $x \in X$ , we have

$$ \begin{align*}\pi |_L (\gamma)=f(\gamma.x)^{-1}\sigma(\gamma,x)f(x). \end{align*} $$

Proof Since the cocycle $\sigma $ is maximal, we know that

$$ \begin{align*}\lambda_{\psi',\psi}(\sigma) = \frac{\lVert \psi' \rVert_\infty}{\lVert \psi \rVert_\infty}. \end{align*} $$

Under condition $(\operatorname {\mathrm {\mbox {NCT}}})$ , if we substitute the value of $\lambda _{\psi ',\psi }(\sigma )$ in equation (5), we get

(6) $$ \begin{align} \int_{L \backslash G} \int_X \psi'(\phi(\overline{g}.\eta_1,x),\ldots,\phi(\overline{g}.\eta_{\bullet+1},x))\,d\mu_X(x)\,d\mu(\overline{g})=\frac{\lVert \psi' \rVert_\infty}{\lVert \psi \rVert_\infty}\psi(\eta_1,\ldots,\eta_{\bullet+1}). \end{align} $$

Moreover, by assumption, $\psi $ attains its essential supremum, from which there exist $\hat {\eta }_1,\ldots ,\hat {\eta }_{\bullet+1} \in G/Q$ , such that

(7) $$ \begin{align} \psi(\hat{\eta}_1,\ldots,\hat{\eta}_{\bullet+1})=\lVert \psi \rVert_\infty. \end{align} $$

By Remark 3.20, we can evaluate equation (6) at $\hat {\eta }_1,\ldots ,\hat {\eta }_{\bullet+1} \in G/Q$ . Hence, by equation (7), we have

(8) $$ \begin{align} \int_{L \backslash G} \int_X \psi'(\phi(\overline{g}.\hat{\eta}_1,x),\ldots,\phi(\overline{g}.\hat{\eta}_{\bullet+1},x))\,d\mu_X(x)\,d\mu(\overline{g})=\lVert \psi' \rVert_\infty. \end{align} $$

This shows that

$$ \begin{align*}\psi'(\phi(\overline{g}.\hat{\eta}_1,x),\ldots,\phi(\overline{g}.\hat{\eta}_{\bullet+1},x))=\lVert \psi' \rVert_\infty \ , \end{align*} $$

for almost every $\overline {g} \in L \backslash G$ and almost every $x \in X$ . Additionally, the $\sigma $ -equivariance of $\phi $ implies that, in fact,

(9) $$ \begin{align} \psi'(\phi(g.\hat{\eta}_1,x),\ldots,\phi(g.\hat{\eta}_{\bullet+1},x))=\lVert \psi' \rVert_\infty \end{align} $$

holds for almost every $g \in G$ and almost every $x \in X$ .

We can define for almost every $x \in \, X$ a map

$$ \begin{align*}\phi_x \colon G/Q \rightarrow Y, \quad \phi_x(\eta):=\phi(\eta,x),\end{align*} $$

which is measurable [Reference Fisher, Morris and WhyteFMW04, Lemma 2.6] and maximal by equation (9). Hence, by the assumptions of Setup 3.27, for almost every $x \in X$ , there must exist an element $g_x \in G'$ , such that

$$ \begin{align*}\phi_x(\eta)=g_x\Pi(\eta), \end{align*} $$

for almost every $\eta \in G/Q$ . This shows that $\phi _x$ lies in the $G'$ -orbit of $\Pi $ . In this way, we get a map

$$ \begin{align*}\widehat{\phi} \colon X \rightarrow G'.\Pi, \quad\widehat{\phi}(x)=\phi_x, \end{align*} $$

which is measurable [Reference Fisher, Morris and WhyteFMW04, Lemma 2.6]. By Setup 3.27, the stabilizer of $\Pi $ is trivial and hence the orbit $G'.\Pi $ is naturally homeomorphic to $G'$ through a map $\jmath \colon G'.\Pi \rightarrow G'$ . Composing the identification $\jmath $ with the map $\widehat {\phi }$ , we get a map

$$ \begin{align*}f \colon X \rightarrow G', \quad f(x):= (\jmath \circ \widehat{\phi})(x), \end{align*} $$

which is defined almost everywhere and it is measurable being the composition of measurable maps (notice that the composition above gives back the element $g_x$ ).

We can now conclude the proof (compare with [Reference Bader, Furman and SauerBFS13b, Proposition 3.2]). Given $\gamma \in L$ , on the one hand, we have

$$ \begin{align*}\phi(\gamma.\eta,\gamma.x)=\sigma(\gamma,x)\phi(\eta,x)=\sigma(\gamma,x)f(x)\Pi(\eta), \end{align*} $$

and on the other,

$$ \begin{align*}\phi(\gamma.\eta,\gamma.x)=f(\gamma.x)\Pi(\gamma.x)=f(\gamma.x)\pi |_L (\gamma)\Pi(\eta). \end{align*} $$

In the second equality, we used the $\pi $ -equivariance of the map $\Pi $ . The fact that $\mathrm{Stab}_{G'}(\Pi )$ is trivial implies that

$$ \begin{align*}\pi |_L (\gamma)=f(\gamma.x)^{-1}\sigma(\gamma,x)f(x), \end{align*} $$

which finishes the proof.

3.5 Applications of the multiplicative formula

For the convenience of the reader, we collect here some examples of applications of Proposition 1.1.

Example 3.30. Let $n \geq 3$ . Let $L \leq G = \operatorname {\mathrm{PO}}^\circ (n, 1)$ be a torsion-free non-uniform lattice and $(X, \mu _X)$ be a standard Borel probability L-space. Following the notation of Setup 3.1, we set $G' = \operatorname {\mathrm{PO}}^\circ (n, 1)$ and $Y = G/Q = \partial \mathbb {H}^n_{\mathbb {R}} \cong \mathbb {S}^{n-1}$ , where Q is a (minimal) parabolic subgroup of G. Using bounded cohomology theory [Reference GromovGro82, Reference Frigerio and MoraschiniFM], one can define the volume $\operatorname {\mathrm {Vol}}(\sigma )$ of a measurable cocycle $\sigma \colon L \times X \to \operatorname {\mathrm{PO}}^\circ (n,1)$ [Reference Moraschini and SaviniMS, §4.1]. As proved by the authors [Reference Moraschini and SaviniMS, Proposition 2], in this setting, the multiplicative constant is given by

$$ \begin{align*}\lambda_{\psi', \psi}(\sigma) = \frac{\operatorname{\mathrm{Vol}}(\sigma)}{\operatorname{\mathrm{Vol}}(L \backslash \mathbb{H}^n)}. \end{align*} $$

Since condition $(\operatorname {\mathrm {\mbox {NCT}}})$ is satisfied for twisted real coefficients [Reference Bucher, Burger, Iozzi and PicardelloBBI13, Lemma 2.2], Proposition 3.25 shows that the following Milnor–Wood inequality holds [Reference Moraschini and SaviniMS, Proposition 15]:

$$ \begin{align*}|\! \operatorname{\mathrm{Vol}}(\sigma) | \leq \operatorname{\mathrm{Vol}}(L \backslash \mathbb{H}^n). \end{align*} $$

Finally one can apply Theorem 3.28 to show that if $\sigma $ is maximal, then $\sigma $ is cohomologous to the cocycle associated with the standard lattice embedding $L \rightarrow G$ . In fact, one can strengthen this result. A cocycle is maximal if and only if it is cohomologous to the cocycle associated with the standard lattice embedding [Reference Moraschini and SaviniMS, Theorem 1].

Similarly, one can apply an analogous strategy for studying the case of closed surfaces. The main difference is that we have to to fix a hyperbolization. Then, maximal cocycles will be cohomologous to the given hyperbolization [Reference Moraschini and SaviniMS, Theorem 5].

Example 3.31. Fix a torsion-free lattice $L \leq G = \mathrm{PSL}(2,\mathbb {C})$ together with a standard Borel probability L-space $(X, \mu _X)$ . Following the notation of Setup 3.1, we set $G' = \mathrm{PSL}(n,\mathbb {C})$ , $Y=\mathscr {F}(n,\mathbb {C})$ is the space of full flags, and $G/Q = \mathbb {P}^1(\mathbb {C})$ . Here, Q is a (minimal) parabolic subgroup of G. The second author defined the Borel invariant $\beta _n(\sigma )$ of a measurable cocycle $\sigma \colon L \times X \to \mathrm{PSL}(n,\mathbb {C})$ [Reference SaviniSava, §4]. Then, the multiplicative constant is given by [Reference SaviniSava, Proposition 1.2]

$$ \begin{align*}\lambda_{\psi', \psi}(\sigma) = \frac{\beta_n(\sigma)}{\operatorname{\mathrm{Vol}}(L \backslash \mathbb{H}^3)}. \end{align*} $$

Since condition $(\operatorname {\mathrm {\mbox {NCT}}})$ is satisfied, Proposition 3.25 leads to the following Milnor–Wood inequality [Reference SaviniSava, Proposition 4.5]:

$$ \begin{align*}| \beta_n(\sigma) | \leq {n+1 \choose 3}\operatorname{\mathrm{Vol}}(L \backslash \mathbb{H}^3). \end{align*} $$

Finally, one can apply Theorem 3.28 to show that if $\sigma $ is maximal, then $\sigma $ is cohomologous to the cocycle associated with the standard lattice embedding $L \rightarrow G$ composed with the irreducible representation $\pi _n:\mathrm{PSL}(2,\mathbb {C}) \rightarrow \mathrm{PSL}(n,\mathbb {C})$ . In fact, also the converse holds true [Reference SaviniSava, Theorem 1.1].