Continuous bounded cohomology and applications to rigidity theory

Abstract

We present a theory of continuous bounded cohomology of locally compact groups with coefficients in Banach modules. A central role is played by amenable actions, as they give rise to relatively injective resolutions. Further, we propose a substitute for the Mautner property, based on the virtual subgroup viewpoint, and we show (Theorem 6) that all compactly generated locally compact groups, e.g. finitely generated groups, satisfy it. This, together with the cohomological characterization of amenable actions, leads to a refined version of a higher degree Lyndon-Hochschild-Serre exact sequence (Theorem 13), which entails a stronger Kunneth type formula for continuous bounded cohomology in degree two. We apply this theory to general irreducible lattices in products of lo- cally compact groups: we obtain notably super-rigidity results for bounded cocycles (Theorem 16 and Corollary 23), rigidity results for actions by diffeomorphisms on the circle (Corollary 22) and vanishing of the stable commutator length (Corollary 32). More applications will be published elsewhere. In the spirit of relative homological algebra, we give for a locally com- pact second countable group G a functorial characterization of the contin- uous bounded cohomology of G with coefficients. The resolutions and the notion of relatively injective objects (Defini- tion 1.4.2) are set up in the category of continuous Banach G-modules, while the coefficients are mainly duals of separable continuous Banach G- modules (henceforth called coefficient modules), including notably separa- ble continuous unitary representations, L ∞ spaces and trivial coefficients. We emphasize that on all Banach G-modules, the G-action is isometric. If E is a coefficient module and S a regular measure G-space (see Defi- nition 1.3.1), let L ∞∗ (S, E) be the space of weak-* measurable essentially bounded maps; we consider the resolution 0 E d L ∞∗(S, E) d L ∞∗ (S 2 ,E ) d L ∞∗ (S 3 ,E ) d ···