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Using a probabilistic argument we show that the second bounded cohomology of an acylindrically hyperbolic group $G$ (e.g., a non-elementary hyperbolic or relatively hyperbolic group, non-exceptional mapping class group, ${rm Out}(F_n)$, dots) embeds via the natural restriction maps into the inverse limit of the second bounded cohomologies of its virtually free subgroups, and in fact even into the inverse limit of the second bounded cohomologies of its hyperbolically embedded virtually free subgroups. This result is new and non-trivial even in the case where $G$ is a (non-free) hyperbolic group. The corresponding statement fails in general for the third bounded cohomology, even for surface groups.
This paper is devoted to the computation of the space $H_b^2(Gamma,H;mathbb{R})$, where $Gamma$ is a free group of finite rank $ngeq 2$ and $H$ is a subgroup of finite rank. More precisely we prove that $H$ has infinite index in $Gamma$ if and only i
Let G be a group, H a hyperbolically embedded subgroup of G, V a normed G-module, U an H-invariant submodule of V. We propose a general construction which allows to extend 1-quasi-cocycles on H with values in U to 1-quasi-cocycles on G with values in
We abstract the notion of an A/QI triple from a number of examples in geometric group theory. Such a triple (G,X,H) consists of a group G acting on a Gromov hyperbolic space X, acylindrically along a finitely generated subgroup H which is quasi-isome
It is proved that the continuous bounded cohomology of SL_2(k) vanishes in all positive degrees whenever k is a non-Archimedean local field. This holds more generally for boundary-transitive groups of tree automorphisms and implies low degree vanishing for SL_2 over S-integers.
We show that for acylindrically hyperbolic groups $Gamma$ (with no nontrivial finite normal subgroups) and arbitrary unitary representation $rho$ of $Gamma$ in a (nonzero) uniformly convex Banach space the vector space $H^2_b(Gamma;rho)$ is infinite