Amenability and Acyclicity in Bounded Cohomology Theory

Johnson's characterization of amenable groups states that a discrete group $\Gamma$ is amenable if and only if $H_b^{n \geq 1}(\Gamma; V) = 0$ for all dual normed $\mathbb{R}[\Gamma]$-modules V. In this paper, we extend the previous result to homomorphisms by proving the converse of the Mapping Theorem: a surjective group homomorphism $\phi \colon \Gamma \to K$ has amenable kernel H if and only if the induced inflation map $H^\bullet_b(K; V^H) \to H^\bullet_b(\Gamma; V)$ is an isometric isomorphism for every dual normed $\mathbb{R}[\Gamma]$-module V. In addition, we obtain an analogous characterization for the (smaller) class of surjective group homomorphisms $\phi \colon \Gamma \to K$ with the property that the inflation maps in bounded cohomology are isometric isomorphisms for all Banach $\Gamma$-modules. Finally, we also prove a characterization of the (larger) class of boundedly acyclic homomorphisms, that is, the class of group homomorphisms $\phi \colon \Gamma \to K$ for which the restriction maps in bounded cohomology $H^\bullet_b(K; V) \to H^\bullet_b(\Gamma; \phi^{-1}V)$ are isomorphisms for a suitable family of dual normed $\mathbb{R}[K]$-modules V including the trivial $\mathbb{R}[K]$-module $\mathbb{R}$. We then extend the first and third results to topological spaces and obtain characterizations of amenable maps and boundedly acyclic maps in terms of the vanishing of the bounded cohomology of their homotopy fibers with respect to appropriate choices of coefficients.