Induced quasi-cocycles on groups with hyperbolically embedded subgroups

TL;DR

This paper introduces a method to extend quasi-cocycles from hyperbolically embedded subgroups to the entire group, revealing infinite-dimensional bounded cohomology and properties related to stable commutator length.

Contribution

It provides a general construction for extending 1-quasi-cocycles from hyperbolically embedded subgroups to the whole group, unifying many previous results.

Findings

Infinite-dimensional second bounded cohomology for groups with hyperbolically embedded subgroups

Hyperbolically embedded subgroups are undistorted in stable commutator length

Uniform approach to known results in hyperbolic and bounded cohomology theory

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-quasi-cocycles on H with values in U to 1-quasi-cocycles on G with values in V. As an application, we show that every group G with a non-degenerate hyperbolically embedded subgroup has dim H^2_b (G, l^p(G))=\infty for p\in [1, \infty). 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.