Geometria Complessa e Geometria Differenziale
Geometria Complessa e Geometria Differenziale
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F. Fournier-Facio - C. Loeh - M. Moraschini

Bounded cohomology and binate groups

created by moraschini on 09 Nov 2021
modified on 11 May 2022


Published Online

Inserted: 9 nov 2021
Last Updated: 11 may 2022

Journal: J. Aust. Math. Soc.
Year: 2022
Doi: 10.1017/S1446788722000106

ArXiv: 2111.04305 PDF


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 (L\"oh). 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.

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