Bounded cohomology and binate groups

Abstract

A group is boundedly acyclic if its bounded cohomology with trivial real coefficients vanishes in all positive degrees. Amenable groups are boundedly acyclic, while the first non-amenable examples were the group of compactly supported homeomorphisms of ${\mathbb{R}}^n$ (Matsumoto--Morita) and mitotic groups (Löh). We prove that binate (alias pseudo-mitotic) groups are boundedly acyclic, which provides a unifying approach to the aforementioned results. Moreover, we show that binate groups are universally boundedly acyclic. We obtain several new examples of boundedly acyclic groups as well as computations of the bounded cohomology of certain groups acting on the circle. In particular, we discuss how these results suggest that the bounded cohomology of the Thompson groups $F$, $T$, and $V$ is as simple as possible.

33 pages, one figure; v3: refs updated and minor changes. The new paragraph 3.1.4 contains examples of amenable binate groups. To appear in J. Aust. Math. Soc

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• On co-amenability for groups and von Neumann algebras
• Leray theorems in bounded cohomology theory
• Amenability and Acyclicity in Bounded Cohomology Theory
• Bounded cohomology of classifying spaces for families of subgroups
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• Compact and finite-type support in the homology of big mapping class groups
• Median quasimorphisms on CAT(0) cube complexes and their cup products
• The space of non-extendable quasimorphisms