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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 if $H_b^2(Gamma,H;mathbb{R})$ is not trivial, and furthermore, if and only if there is an isometric embedding $oplus_infty^nmathcal{D}(mathbb{Z})hookrightarrow H_b^2(Gamma,H;mathbb{R})$, where $mathcal{D}(mathbb{Z})$ is the space of bounded alternating functions on $mathbb{Z}$ equipped with the defect norm.
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
Given a group acting on a Gromov hyperbolic space, Bestvina and Fujiwara introduced the WPD property --- weak proper discontinuity --- for studying the 2nd bounded cohomology of the group. We carry out a more general study of second bounded cohomolog
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
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 give explicit necessary and sufficient conditions for the abstract commensurability of certain families of 1-ended, hyperbolic groups, namely right-angled Coxeter groups defined by generalized theta-graphs and cycles of generalized theta-graphs, a