Induced quasicocycles on groups with hyperbolically embedded subgroups

Hull, Michael; Osin, Denis

doi:10.2140/agt.2013.13.2635

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Induced quasicocycles on groups with hyperbolically embedded subgroups

Michael Hull and Denis Osin

Algebraic & Geometric Topology 13 (2013) 2635–2665

DOI: 10.2140/agt.2013.13.2635

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 $dim H_{b}^{2} (G, ℓ^{p} (G)) = \infty$ for $p \geq 1$ . 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

References

Publication

Received: 26 May 2012

Revised: 28 January 2013

Accepted: 9 February 2013

Published: 7 July 2013

Authors

Michael Hull
Department of Mathematics Vanderbilt University 1326 Stevenson Center Nashville, TN 37240 USA
http://math.vanderbilt.edu/people/hull
Denis Osin
Mathematics Department Vanderbilt University 1326 Stevenson Center Nashville, TN 37240 USA
http://math.vanderbilt.edu/people/osin

Volume 13, issue 5 (2013)