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Induced quasicocycles on groups with hyperbolically embedded subgroups

Michael Hull and Denis Osin

Algebraic & Geometric Topology 13 (2013) 2635–2665

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 dimHb2(G,p(G)) = for p 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.

hyperbolic space, hyperbolically embedded subgroups, left regular representation, quasicocycle, bounded cohomology, stable commutator length
Mathematical Subject Classification 2010
Primary: 20F65, 20F67, 20J06, 43A15, 57M07
Received: 26 May 2012
Revised: 28 January 2013
Accepted: 9 February 2013
Published: 7 July 2013
Michael Hull
Department of Mathematics
Vanderbilt University
1326 Stevenson Center
Nashville, TN 37240
Denis Osin
Mathematics Department
Vanderbilt University
1326 Stevenson Center
Nashville, TN 37240