#### Volume 13, issue 5 (2013)

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Induced quasicocycles on groups with hyperbolically embedded subgroups

### Michael Hull and Denis Osin

Algebraic & Geometric Topology 13 (2013) 2635–2665
##### Abstract

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 $1$–quasicocycles on $H$ with values in $U$ to $1$–quasicocycles on $G$ with values in $V$. As an application, we show that every group $G$ with a nondegenerate hyperbolically embedded subgroup has $dim{H}_{b}^{2}\left(G,{\ell }^{p}\left(G\right)\right)=\infty$ for $p\ge 1$. This covers many previously known results in a uniform way. Applying our extension to quasimorphisms and using Bavard duality, we also show that hyperbolically embedded subgroups are undistorted with respect to the stable commutator length.

##### Keywords
hyperbolic space, hyperbolically embedded subgroups, left regular representation, quasicocycle, bounded cohomology, stable commutator length
##### Mathematical Subject Classification 2010
Primary: 20F65, 20F67, 20J06, 43A15, 57M07