Induced quasicocycles on groups with hyperbolically embedded subgroups

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.