Integrable measure equivalence and rigidity of hyperbolic lattices

Abstract

We study rigidity properties of lattices in $\operatorname {Isom}(\mathbf {H}^{n})\simeq \mathrm {SO}_{n,1}({\mathbb{R}})$, n≥3, and of surface groups in $\operatorname {Isom}(\mathbf {H}^{2})\simeq \mathrm {SL}_{2}({\mathbb{R}})$ in the context of integrable measure equivalence. The results for lattices in $\operatorname {Isom}(\mathbf {H}^{n})$, n≥3, are generalizations of Mostow rigidity; they include a cocycle version of strong rigidity and an integrable measure equivalence classification. Despite the lack of Mostow rigidity for n=2 we show that cocompact lattices in $\operatorname {Isom}(\mathbf {H}^{2})$ allow a similar integrable measure equivalence classification.

Appendices

Appendix A: Measure equivalence

The appendix contains some general facts related to measure equivalence (Definition 1.1), the strong ICC property (Definition 2.2), and the notions of taut couplings and groups (Definition 1.3).

1.1 A.1 The category of couplings

Measure equivalence is an equivalence relation on unimodular lcsc groups. Let us describe explicitly the constructions which show reflexivity, symmetry and transitivity of measure equivalence.

1.1.1 A.1.1 Tautological coupling

The tautological coupling is the (G×G)-coupling (G,m _G ) given by $(g_{1},g_{2}):g\mapsto g_{1} g g_{2}^{-1}$. It demonstrates reflexivity of measure equivalence.

1.1.2 A.1.2 Duality

Symmetry is implied by the following: Given a (G,H)-coupling (Ω,m) the dual (Ω ^∗,m ^∗) is the (H,G)-coupling Ω ^∗ with the same underlying measure space (Ω,m) and the H×G-action (h,g):ω ^∗↦(g,h)ω ^∗.

1.1.3 A.1.3 Composition of couplings

Compositions defined below shows that measure equivalence is a transitive relation. Let G ₁,H,G ₂ be unimodular lcsc groups, and (Ω _i ,m _i ) be a (G _i ,H)-coupling for i∈{1,2}. We describe the (G ₁,G ₂)-coupling $\varOmega_{1}\times_{H}{\varOmega}^{*}_{2}$ modeled on the space of H-orbits on (Ω ₁×Ω ₂,m ₁×m ₂) with respect to the diagonal H-action. Consider measure isomorphisms for (Ω _i ,mi) as in (1.1): For i∈{1,2} there are finite measure spaces (X _i ,μ _i ) and (Y _i ,ν _i ), measure-preserving actions G _i ↷(X _i ,μ _i ) and H↷(Y _i ,ν _i ), measurable cocycles α _i :G _i ×X _i →H and β _i :H×Y _i →G _i , and measure space isomorphisms G _i ×Y _i ≅Ω _i ≅H×X _i with respect to which the G _i ×H-actions are given by

The space $\varOmega_{1}\times_{H}{\varOmega}^{*}_{2}$ with its natural G ₁×G ₂-action is equivariantly isomorphic to (X ₁×X ₂×H,μ ₁×μ ₂×m _H ) endowed with the G ₁×G ₂-action

$$(g_1,g_2): (x_1,x_2,h)\mapsto \bigl(g_1\mathbin{.}x_1,\, g_2\mathbin{.}x_2,\, \alpha_1(g_1,x_1) h \alpha_2(g_2,x_2)^{-1} \bigr). $$

To see that it is a (G ₁,G ₂)-coupling, we identify this space with Z×G ₁ equipped with the action

$$(g_1,g_2): \bigl(g',z\bigr)\mapsto \bigl(g_1 g' c(g_2,z)^{-1}, g_2\mathbin{.}z\bigr)\quad\bigl(g'\in G_1,\ z\in Z \bigr) $$

where Z=X ₂×Y ₁, while the action G ₂↷Z and the cocycle c:G ₂×Z→G ₁ are given by

$$ \begin{aligned} &g_2:(x,y) \mapsto\bigl(g_2\mathbin{.}x,\alpha_2(g_2,x)\mathbin{.}y\bigr), \\ &c\bigl(g_2,(x,y)\bigr)=\beta_1\bigl( \alpha_2(g_2,x), y\bigr). \end{aligned} $$

(A.1)

Similarly, $\varOmega_{1}\times_{H}{\varOmega}^{*}_{2}\cong W\times G_{2}$, for W≅X ₁×Y ₂.

1.1.4 A.1.4 Morphisms

Let (Ω _i ,m _i ), i∈{1,2}, be two (G,H)-couplings. Let F:Ω ₁→Ω ₂ be a measurable map such that for m ₁-a.e. ω∈Ω ₁ and every g∈G and every h∈H

$$F\bigl((g,h)\omega\bigr)=(g,h)F(\omega). $$

Such maps are called quotient maps or morphisms.

1.1.5 A.1.5 Compact kernels

Let (Ω,m) be a (G,H)-coupling, and let

$$\{1\}\to K\to G\to\bar{G}\to\{1\} $$

be a short exact sequence where K is compact. Then the natural quotient space $(\bar{\varOmega},\bar{m})=(\varOmega,m)/K$ is a $(\bar{G},H)$-coupling, and the natural map $F:\varOmega\to\bar {\varOmega}$, F:ω↦Kω, is equivariant in the sense of $F((g,h)\omega)=(\bar{g},h) F(\omega)$. This may be considered as an isomorphism of couplings up to compact kernel.

1.1.6 A.1.6 Passage to lattices

Let (Ω,m) be a (G,H)-coupling, and let Γ<G be a lattice. By restricting the G×H-action on (Ω,m) to Γ×H we obtain a (Γ,H)-coupling. Formally, this follows by considering (G,m _G ) as a Γ×G-coupling and considering the composition G× _G Ω as Ω with the Γ×H-action.

1.2 A.2 L^p-integrability conditions

Let G and H be compactly generated unimodular lcsc groups equipped with proper norms |⋅| _G and |⋅| _H . Let c:G×X→H be a measurable cocycle, and fix some p∈[1,∞). For g∈G we define

$$\|g\|_{c,p}= \biggl(\int_X \bigl|c(g,x)\bigr|^p_H \,d\mu(x) \biggr)^{1/p}. $$

For p=∞ we use the essential supremum. Assume that ∥g∥ _c,p <∞ for a.e. g∈G. We claim that there are constants a,A>0 so that for every g∈G

$$ \|g\|_{c,p}\le A\cdot|g|_G+a. $$

(A.2)

Hence c is L^p-integrable in the sense of Definition 1.5. The key observation here is that ∥−∥ _c,p is subadditive. Indeed, by the cocycle identity, subadditivity of the norm |−| _H , and the Minkowski inequality, for any g ₁,g ₂∈G we get

For t>0 denote E _t ={g∈G:∥g∥ _c,p <t}. We have E _t ⋅E _s ⊆E _s+t for any t,s>0. Fix t large enough so that m _G (E _t )>0. By [11, Corollary 12.4 on p. 235], E _2t ⊇E _t ⋅E _t has a non-empty interior. Hence any compact subset of G can be covered by finitely many translates of E _2t . The subadditivity implies that ∥g∥ _c,p is bounded on compact sets. This gives (A.2).

Lemma A.1

Let G,H,L be compactly generated groups, G↷(X,μ), H↷(Y,ν) be finite measure-preserving actions, and α:G×X→H and β:H×Y→L be L^p-integrable cocycles for some 1≤p≤∞. Consider Z=X×Y and G↷Z by g:(x,y)↦(g.x,α(g,x).y). Then the cocycle γ:G×Z→L given by

$$\gamma\bigl(g,(x,y)\bigr)=\beta\bigl(\alpha(g,x),y\bigr). $$

is L^p-integrable.

Proof

For p=∞ the claim is obvious. Assume p<∞. Let A,a,B,b be constants such that ∥h∥ _β,p ≤B⋅|h| _H +b and ∥g∥ _α,p ≤A⋅|g| _G +a. Then

for appropriate constants c>0 and C>0. □

Lemma A.2

Let G ₁,H,G ₂ be compactly generated unimodular lcsc groups. For i∈{1,2} let (Ω _i ,m _i ) be an L^p-integrable (G _i ,H)-coupling. Then $\varOmega_{1}\times_{H} {\varOmega}^{*}_{2}$ is an L^p-integrable (G ₁,G ₂)-coupling.

Proof

This follows from Lemma A.1 using the explicit description (A.1) of the cocycles for $\varOmega_{1}\times_{H} {\varOmega}^{*}_{2}$. □

We conclude that for each 1≤p≤∞, L^p-measure equivalence is an equivalence relation between compactly generated unimodular lcsc groups.

1.3 A.3 Tautening maps

Lemma A.3

Let G be a lcsc group, Γ a countable group and j ₁,j ₂:Γ→G be homomorphisms with Γ _i =j _i (Γ) being lattices in G. Assume that G is taut (resp. p-taut and Γ _i are L^p-integrable). Then there exists g∈G so that

$$j_2(\gamma)=g\,j_1(\gamma)\,g^{-1}\quad( \gamma\in\varGamma). $$

If $\pi:G\to \mathcal {G}$ is a continuous homomorphism into a Polish group and G is taut relative to $\pi:G\to \mathcal {G}$ (resp. G is p-taut relative to $\pi:G\to \mathcal {G}$ and Γ _i are L^p-integrable) then there exists $y\in \mathcal {G}$ with

$$\pi\bigl(j_2(\gamma)\bigr)=y \pi\bigl(j_1(\gamma) \bigr)y^{-1}\quad(\gamma\in \varGamma). $$

Proof

We prove the more general second statement. The group

$$\varDelta =\bigl\{\bigl(j_1(\gamma),j_2(\gamma)\bigr)\in G \times G \mid\gamma\in \varGamma\bigr\} $$

is a closed discrete subgroup in G×G. The G×G-space Ω=(G×G)/Δ equipped with the G×G-invariant measure is easily seen to be a (G,G)-coupling. It will be L^p-integrable if Γ ₁ and Γ ₂ are L^p-integrable lattices. Let $\varPhi:\varOmega\to \mathcal {G}$ be the tautening map. There are a,b∈G and $x\in \mathcal {G}$ such that for all g ₁,g ₂∈G

$$\varPhi\bigl((g_1a,g_2b)\varDelta _f\bigr)= \pi(g_1) x \pi(g_2)^{-1}. $$

Since (a,b) and (j ₁(γ)^a  a, j ₂(γ)^b  b) are in the same Δ-coset, where g ^h=hgh ⁻¹, we get for all g ₁,g ₂∈G and every γ∈Γ

$$\pi(g_1) x \pi(g_2)^{-1}=\pi(g_1) \pi\bigl(j_1(\gamma)^a\bigr) x\pi \bigl(j_2( \gamma)^b\bigr)^{-1} \pi(g_2)^{-1}. $$

This implies that j ₁ and j ₂ are conjugate homomorphisms. □

The following lemma relates tautening maps Φ:Ω→G and cocycle rigidity for ME-cocycles.

Lemma A.4

Let G be a unimodular lcsc group, $\mathcal {G}$ be a Polish group, $\pi :G\to \mathcal {G}$ a continuous homomorphism. Let (Ω,m) be a (G,G)-coupling and α:G×X→G, β:G×Y→G be the corresponding ME-cocycles. Then there is a tautening map $\varOmega\to \mathcal {G}$ iff the $\mathcal {G}$-valued cocycle π∘α is conjugate to π, that is,

$$\pi\circ\alpha(g,x)=f(g\mathbin{.}x)^{-1} \pi(g) f(x). $$

Moreover, Ω is taut relative to π if such a measurable map $f:X\to \mathcal {G}$ is unique. This is also equivalent to π∘β being uniquely conjugate to $\pi: G\to \mathcal {G}$.

Proof

Let α:G×X→G be the ME-cocycle associated to a measure space isomorphism i:(G,m _G )×(X,μ)→(Ω,m) as in (1.1). In particular,

$$(g_1,g_2): i(g,x)\mapsto i\bigl(g_2 g \alpha(g_1,x)^{-1},\,g_1\mathbin{.}x\bigr). $$

We shall now establish a 1-to-1 correspondence between Borel maps $f:X\to\nobreak \mathcal {G}$ with

$$\pi\circ\alpha(g,x)=f(g\mathbin{.}x)^{-1}\pi(g) f(x) $$

and tautening maps $\varPhi:\varOmega\to \mathcal {G}$. Given f as above one verifies that

$$\varPhi:\varOmega\to \mathcal {G},\qquad\varPhi\bigl(i(g,x)\bigr)=f(x) \pi(g)^{-1} $$

is G×G-equivariant.

For the converse direction, suppose Φ:Ω→G is a tautening map. Thus,

$$g_1 \varPhi(g_0,x) g_2^{-1}= \varPhi\bigl((g_1,g_2) (g_0,x)\bigr)=\varPhi \bigl(g_2 g_0 \alpha(g_1,x)^{-1},\, g_1\mathbin{.}x\bigr). $$

For μ-a.e. x∈X and a.e. g∈G the value of Φ(g,x)g is constant f(x). If we substitute g ₀=g ₁=g and g ₂=gα(g,x)g ⁻¹ in the above identity, then we obtain α(g,x)=f(g.x)⁻¹ gf(x). □

Lemma A.5

Let G be a unimodular lcsc group, $\mathcal {G}$ be a Polish group, $\pi :G\to \mathcal {G}$ a continuous homomorphism. Then G is (p-)taut, that is every (p-integrable) (G,G)-coupling is taut relative to π, iff every ergodic (p-integrable) (G,G)-coupling is taut relative to π.

Proof

We give the proof in the p-integrable case, the case without the integrability condition being simpler. We assume that every ergodic p-integrable (G,G)-coupling is taut relative to π and let (Ω,m) be an arbitrary p-integrable coupling. We fix a fundamental domain (X,μ) for the second G action such that the associated cocycle α:G×X→G is p-integrable.

Let μ=∫μ _t   dη(t) be the G-ergodic decomposition of (X,μ). By [15, Lemma 2.2] it corresponds to the (G×G)-ergodic decomposition of (Ω,m) into ergodic couplings (Ω,m _t ), so that m=∫m _t   dη(t).

Let |_|:G→ℕ be the length function associated to some word-metric on G. The p-integrability of α means that

$$\int\int_X{ \bigl\vert\alpha(g,x) \bigr\vert}^p\,d \mu_t(x)\,d\eta (t)=\int_X{ \bigl\vert\alpha(g,x) \bigr\vert}^p\,d\mu(x)<\infty $$

for every g∈G. By Fubini’s theorem ∫ _X |α(g,x)|^p  dμ _t (x)<∞ for η-a.e. t. Hence (Ω,m _t ) is p-integrable for η-a.e. m _t , and in particular it is taut relative to π, by our assumption. It follows by Lemma A.4 that the cocycle π∘α is conjugate to the constant cocycle π over η-a.e. (X,μ _t ). Then by [13, Corollary 3.6]¹⁰ π∘α is conjugate to π over (X,μ). Again, by Lemma A.4 we conclude that (Ω,m) is taut relative to π. □

1.4 A.4 Strong ICC property

Lemma A.6

Let G be a unimodular lcsc group, $\mathcal {G}$ a Polish group, $\pi:G\to \mathcal {G}$ a continuous homomorphism. Let Γ<G be a lattice. Then $\mathcal {G}$ is strongly ICC relative to π(G) if and only if it is strongly ICC relative to π(Γ).

Proof

Clearly if $\mathcal {G}$ is strongly ICC relative to π(Γ) then it is also strongly ICC relative to π(G). The reverse implication follows by averaging a π(Γ) invariant measure over G/Γ. □

Lemma A.7

Let G be a unimodular lcsc group, $\mathcal {G}$ a Polish group, $\pi:G\to \mathcal {G}$ a continuous homomorphism. Suppose that $\mathcal {G}$ is not strongly ICC relative to π(G). Then there is a (G,G)-coupling (Ω,m) with two distinct tautening maps to $\mathcal {G}$.

Proof

Let μ be a Borel probability measure on $\mathcal {G}$ invariant under conjugations by π(G). Consider $\varOmega=G\times \mathcal {G}$ with the measure m=m _G ×μ where m _G denotes the Haar measure, and measure-preserving G×G-action

$$(g_1,g_2): (g,x)\mapsto\bigl(g_1 g g_2^{-1},\, \pi(g_2)\, x\, \pi(g_2)^{-1}\bigr). $$

This is clearly a (G,G)-coupling and the following measurable maps $\varPhi_{i}:\varOmega\to \mathcal {G}$, i∈{1,2}, are G×G-equivariant: Φ ₁(g,x)=π(g) and Φ ₂(g,x)=π(g)⋅x. Note that Φ ₁=Φ ₂ on a conull set iff μ=δ _e . □

Lemma A.8

Let G be a unimodular lcsc group and $\mathcal {G}$ a Polish group. Assume that $\mathcal {G}$ is strongly ICC relative to π(G). Let (Ω,m) be a (G,G)-coupling. Then:

• (1) There is at most one tautening map $\varPhi\colon\varOmega\to \mathcal {G}$.

• (2) Let F:(Ω,m)→(Ω ₀,m ₀) be a morphism of (G,G)-couplings and suppose that there exists a tautening map $\varPhi:\varOmega\to \mathcal {G}$. Then it descends to Ω ₀, i.e., Φ=Φ ₀∘F for a unique tautening map $\varPhi_{0}:\varOmega_{0}\to \mathcal {G}$.

• (3) If Γ ₁,Γ ₂<G are lattices, then $\varPhi:\varOmega \to \mathcal {G}$ is unique as a Γ ₁×Γ ₂-equivariant map.

• (4) If Γ ₁,Γ ₂<G are lattices, and (Ω,m) admits a Γ ₁×Γ ₂-equivariant map $\varPhi:\varOmega \to \mathcal {G}$, then Φ is G×G-equivariant.

• (5) If $\eta:\varOmega\to \operatorname {Prob}(\mathcal {G})$, ω↦η _ω , is a measurable G×G-equivariant map to the space of Borel probability measures on $\mathcal {G}$ endowed with the weak topology, then it takes values in Dirac measures: We have η _ω =δ _Φ(ω), where $\varPhi:\varOmega\to \mathcal {G}$ is the unique tautening map.

Proof

We start from the last claim and deduce the other ones from it.

(5). Given an equivariant map $\eta:\varOmega\to \operatorname {Prob}(\mathcal {G})$ consider the convolution

$$\nu_\omega=\check{\eta}_\omega* \eta_\omega, $$

namely the image of η _ω ×η _ω under the map (a,b)↦a ⁻¹⋅b. Then

$$\nu_{(g,h)\omega}=\nu_\omega^{\pi(g)}\quad(g,h\in G), $$

where the latter denotes the push-forward of ν _ω under the conjugation

$$a\mapsto a^{\pi(g)}=\pi(g)^{-1}a \pi(g). $$

In particular, the map ω↦ν _ω is invariant under the action of the second G-factor. Therefore ν _ω descends to a measurable map $\tilde{\nu}:\varOmega/G\to \operatorname {Prob}(\mathcal {G})$, satisfying

$$\tilde{\nu}_{g.x}=\tilde{\nu}_x^{\pi(g)}\quad(x\in X= \varOmega/G,\ g\in G). $$

Here we identify Ω/G with a finite measure space (X,μ) as in (1.1). Consider the center of mass

$$\bar{\nu}=\frac{1}{\mu(X)}\int_{X}\tilde{\nu}_x \,d\mu(x). $$

It is a probability measure on $\mathcal {G}$, which is invariant under conjugations. By the strong ICC property relative to π(G) we get $\bar{\nu}=\delta_{e}$. Since δ _e is an extremal point of $\operatorname {Prob}(\mathcal {G})$, it follows that m-a.e. ν(ω)=δ _e . This implies that η _ω =δ _Φ(ω) for some measurable $\varPhi:\varOmega\to \mathcal {G}$. The latter is automatically G×G-equivariant.

(1). If $\varPhi_{1},\varPhi_{2}:\varOmega\to \mathcal {G}$ are tautening maps, then $\eta_{\omega}=\frac{1}{2}(\delta_{\varPhi_{1}(\omega)}+\delta_{\varPhi_{2}(\omega)})$ is an equivariant map $\varOmega\to \operatorname {Prob}(\mathcal {G})$. By (5) it takes values in Dirac measures, which is equivalent to the m-a.e. equality Φ ₁=Φ ₂.

(2). Disintegration of m with respect to m ₀ gives a G×G-equivariant measurable map $\varOmega_{0}\to\mathcal{M}(\varOmega)$, $\omega\mapsto m_{\omega_{0}}$, to the space of finite measures on Ω. Then the map $\eta:\varOmega_{0}\to \operatorname {Prob}(\mathcal {G})$, given by

$$\eta_{\omega_0}=\|m_{\omega_0}\|^{-1}\cdot\varPhi_* (m_{\omega_0}) $$

is G×G-equivariant. Hence by (5), $\eta_{\omega _{0}}=\delta_{\varPhi_{0}(\omega_{0})}$ for the unique tautening map $\varPhi_{0}:\varOmega_{0}\to \mathcal {G}$. The relation Φ=Φ ₀∘F follows from the fact that Dirac measures are extremal.

(3) follows from (4) and (1).

(4). The claim is equivalent to: For m-a.e. ω∈Ω the map $F_{\omega}\colon G\times G\to \mathcal {G}$ with

$$F_\omega(g_1,g_2)=\pi(g_1)^{-1} \,\varPhi\bigl((g_1,g_2)\omega\bigr)\,\pi(g_2) $$

is m _G ×m _G -a.e. constant Φ ₀(ω). Note that the family {F _ω } has the following equivariance property: For g ₁,g ₂,h ₁,h ₂∈G we have

Since Φ is Γ ₁×Γ ₂-equivariant, for m-a.e. ω∈Ω the map F _ω descends to G/Γ ₁×G/Γ ₂. Let $\eta_{\omega}\in \operatorname {Prob}(\mathcal {G})$ denote the distribution of F _ω (⋅,⋅) over the probability space G/Γ ₁×G/Γ ₂, that is, for a Borel subset $E\subset \mathcal {G}$

$$\eta_\omega(E)=m_{G/\varGamma_1}\times m_{G/\varGamma_2}\bigl\{ (g_1,g_2) \mid F_\omega(g_1,g_2) \in E\bigr\}. $$

Since this measure is invariant under translations by G×G, it follows that η _ω is a G×G-equivariant maps $\varOmega\to \operatorname {Prob}(\mathcal {G})$. By (5) one has η _ω =δ _f(ω) for some measurable G×G-equivariant map $f:\varOmega\to \mathcal {G}$. Hence F _ω (g ₁,g ₂)=f(ω) for a.e. g ₁,g ₂∈G; it follows that

$$ \varPhi\bigl((g_1,g_2)\omega\bigr)= \pi(g_1) \varPhi(\omega) \pi(g_2)^{-1} $$

(A.3)

holds for m _G ×m _G ×m-a.e. (g ₁,g ₂,ω). □

Corollary A.9

Let $\pi:G\to \mathcal {G}$ be as above and assume that $\mathcal {G}$ is strongly ICC relative to π(G). Then the collection of all (G,G)-couplings which are taut relative to $\pi:G\to \mathcal {G}$ is closed under the operations of taking the dual, compositions, quotients and extensions.

Proof

The uniqueness of tautening maps follow from the relative strong ICC property (Lemma A.8.(1)). Hence we focus on the existence of such maps.

Let $\varPhi:\varOmega\to \mathcal {G}$ be a tautening map. Then Ψ(ω ^∗)=Φ(ω)⁻¹ is a tautening map ${\varOmega}^{*}\to \mathcal {G}$.

Let $\varPhi_{i}:\varOmega_{i}\to \mathcal {G}$, i=1,2, be tautening maps. Then Ψ([ω ₁,ω ₂])=Φ(ω ₁)⋅Φ(ω ₂) is a tautening map $\varOmega_{1}\times_{G}\varOmega_{2}\to \mathcal {G}$.

If F:(Ω ₁,m ₁)→(Ω ₂,m ₂) is a quotient map and $\varPhi_{1}:\varOmega_{1}\to \mathcal {G}$ is a tautening map, then, by Lemma A.8.(2), Φ ₁ factors as Φ ₁=Φ ₂∘F for a tautening map $\varPhi_{2}:\varOmega_{2}\to \mathcal {G}$. On the other hand, given a tautening map $\varPhi_{2}:\varOmega_{2}\to \mathcal {G}$, the map Φ ₁=Φ ₂∘F is tautening for Ω ₁. □

Appendix B: Bounded cohomology

Our background references for bounded cohomology, especially for the functorial approach to it, are [6, 41]. We summarize what we need from Burger-Monod’s theory of bounded cohomology. Since we restrict to discrete groups, some results we quote from this theory already go back to Ivanov [31].

2.1 B.1 Banach modules

All Banach spaces are over the field ℝ of real numbers. By the dual of a Banach space we understand the normed topological dual. The dual of a Banach space E is denoted by E ^∗. Let Γ be a discrete and countable group. A Banach Γ-module is a Banach space E endowed with a group homomorphism π from Γ into the group of isometric linear automorphisms of E. We use the module notation γ⋅e=π(γ)(e) or just γe=π(γ)(e) for γ∈Γ and e∈E whenever the action is clear from the context. The submodule of Γ-invariant elements is denoted by E ^Γ. Note that E ^Γ⊂E is closed.

If E and F are Banach Γ-modules, a Γ-morphism E→F is a Γ-equivariant continuous linear map. The space $\mathcal {B}(E,F)$ of continuous, linear maps E→F is endowed with a natural Banach Γ-module structure via

$$ (\gamma\cdot f) (e)=\gamma f\bigl( \gamma^{-1}e\bigr). $$

(B.1)

The contragredient Banach Γ-module structure E ^♯ associated to E is by definition $\mathcal {B}(E,{\mathbb{R}})=E^{\ast}$ with the Γ-action (B.1). A coefficient Γ-module is a Banach Γ-module E contragredient to some separable continuous Banach Γ-module denoted by E ^♭. The choice of E ^♭ is part of the data. The specific choice of E ^♭ defines a weak-∗ topology on E. The only examples that appear in this paper are E=L^∞(X,μ) with E ^♭=L ¹(X,μ) and E=E ^♭=ℝ.

For a coefficient Γ-module E let $\mathrm{C}_{\mathrm {b}}^{k}(\varGamma,E)$ be the Banach Γ-module L^∞(Γ ^k+1,E) consisting of bounded maps from Γ ^k+1 to E endowed with the supremum norm and the Γ-action:

$$ (\gamma\cdot f) (\gamma_0,\dots, \gamma_k)=\gamma\cdot f\bigl(\gamma^{-1} \gamma_0,\dots,\gamma^{-1}\gamma_k\bigr). $$

(B.2)

For a coefficient Γ-module E and a standard Borel Γ-space S with quasi-invariant measure let $\mathrm {L}_{\mathrm{w}\ast}^{\infty}(S, E)$ be the space of weak-∗-measurable essentially bounded maps from S to E, where maps are identified if they only differ on a null set. The space $\mathrm {L}_{\mathrm{w}\ast }^{\infty}(S, E)$ is endowed with the essential supremum norm and the Γ-action (B.2). For a measurable space X the Banach space $\mathcal {B}^{\infty}(X, E)$ is the space of weak-∗-measurable bounded maps from X to E endowed with supremum norm [4, Sect. 2] and the Γ-action (B.2).

2.2 B.2 Injective resolutions

Let Γ be a discrete group and E be a Banach Γ-module. The sequence of Banach Γ-modules $\mathrm{C}_{\mathrm {b}}^{k}(\varGamma, E)$, k≥0, becomes a chain complex of Banach Γ-modules via the standard homogeneous coboundary operator

$$ d(f) (\gamma_0,\dots, \gamma_k)=\sum_{i\ge0}^k(-1)^i f(\gamma_0,\dots, \hat{\gamma_i},\dots, \gamma_k). $$

(B.3)

The bounded cohomology $\mathrm{H}_{\mathrm{b}}^{\bullet}(\varGamma,E)$ of Γ with coefficients E is the cohomology of the complex of Γ-invariants $\mathrm{C}_{\mathrm{b}}^{\bullet}(\varGamma, E)^{\varGamma}$. The bounded cohomology $\mathrm{H}_{\mathrm{b}}^{\bullet}(\varGamma,E)$ inherits a semi-norm from $\mathrm{C}_{\mathrm{b}}^{\bullet}(\varGamma, E)$: The (semi-)norm of an element $x\in\mathrm{H}_{\mathrm {b}}^{k}(\varGamma, E)$ is the infimum of the norms of all cocycles in the cohomology class x.

Next we briefly recall the functorial approach to bounded cohomology as introduced by Ivanov [31] for discrete groups and further developed by Burger-Monod [6, 41]. We refer for the definition of relative injectivity of a Banach Γ-module to [41, Definition 4.1.2 on p. 32]. A strong resolution E ^• of E is a resolution, i.e. an acyclic complex,

$$0\to E\to E^0\to E^1\to E^2\to\cdots $$

of Banach Γ-modules that has a chain contraction which is contracting with respect to the Banach norms. The key to the functorial definition of bounded cohomology are the following two theorems:

Theorem B.1

([6, Proposition 1.5.2])

Let E and F be Banach Γ-modules. Let E ^• be a strong resolution of E. Let F ^• be a resolution F by relatively injective Banach Γ-modules. Then any Γ-morphism E→F extends to a Γ-morphism of resolutions E ^•→F ^• which is unique up to Γ-homotopy. Hence E→F induces functorially continuous linear maps H^•(E ^{• Γ})→H^•(F ^{• Γ}).

Theorem B.2

([41, Corollary 7.4.7 on p. 80])

Let E be a Banach Γ-module. The complex $E\to\mathrm{C}_{\mathrm{b}}^{\bullet}(\varGamma,E)$ with $E\to \mathrm{C}_{\mathrm{b}}^{0}(\varGamma,E)$ being the inclusion of constant functions is a strong, relatively injective resolution.

For a coefficient Γ-module, a measurable space X with measurable Γ-action, and a standard Borel Γ-space S with quasi-invariant measure we obtain chain complexes $\mathcal {B}^{\infty}(X^{\bullet+1},E)$ and L_w∗(S ^•+1,E) of Banach Γ-modules via the standard homogeneous coboundary operators (similar as in (B.3)).

The following result is important for expressing induced maps in bounded cohomology in terms of boundary maps [4].

Proposition B.3

([4, Proposition 2.1])

Let E be a coefficient Γ-module. Let X be a measurable space with measurable Γ-action. The complex $E\to \mathcal {B}^{\infty}(X^{\bullet+1},E)$ with $E\to \mathcal {B}^{\infty}(X,E)$ being the inclusion of constant functions is a strong resolution of E.

The next theorem is one of the main results of the functorial approach to bounded cohomology by Burger-Monod:

Theorem B.4

([6, Corollary 2.3.2, 41, Theorem 7.5.3 on p. 83])

Let S be a regular Γ-space and be E a coefficient Γ-module. Then E→L_w∗(S ^•+1,E) with E→L_w∗(S ^•+1,E) being the inclusion of constant functions is a strong resolution. If, in addition, S is amenable in the sense of Zimmer [41, Definition 5.3.1], then each L_w∗(S ^k+1,E) is relatively injective, and according to Theorem B.1 the cohomology groups H^•(L_w∗(S ^•+1,E)^Γ) are canonically isomorphic to $\mathrm{H}_{\mathrm{b}}^{\bullet}(\varGamma,E)$.

Definition B.5

Let S be a standard Borel Γ-space with a quasi-invariant probability measure μ. Let E be a coefficient Γ-module. The Poisson transform $\mathrm {PT}^{\bullet}: \mathrm {L}_{\mathrm{w}\ast}(S^{\bullet+1},E)\to\mathrm{C}_{\mathrm {b}}^{\bullet}(\varGamma , E)$ is the Γ-morphism of chain complexes defined by

$$\mathrm {PT}^k(f) (\gamma_0,\dots,\gamma_k)= \int _{S^{k+1}} f(\gamma_0 s_0,\dots, \gamma_k s_k)\,d\mu(s_0)\cdots d \mu(s_k). $$

If S is amenable, then the Poisson transform induces a canonical isomorphism in cohomology (Theorem B.4). By the same theorem this isomorphism does not depend on the choice of μ within a given measure class.