Extending higher dimensional quasi-cocycles

Abstract

Let G be a group admitting a non-elementary acylindrical action on a Gromov hyperbolic space (for example, a non-elementary relatively hyperbolic group, or the mapping class group of a closed hyperbolic surface, or Out(F_n) for n>1). We prove that, in degree 3, the bounded cohomology of G with real coefficients is infinite-dimensional. Our proof is based on an extension to higher degrees of a recent result by Hull and Osin. Namely, we prove that, if H is a hyperbolically embedded subgroup of G and V is any G-module, then any n-quasi cocycle on H with values in V may be extended to G. Also, we show that our extensions detect the geometry of the embedding of hyperbolically embedded subgroups, in a suitable sense.

Minor revisions. This version has been accepted for publication by the Journal of Topology

References in corpus (3)

• The Median Class and Superrigidity of Actions on CAT(0) Cube Complexes
• Commensurating endomorphisms of acylindrically hyperbolic groups and applications
• Small cancellation in acylindrically hyperbolic groups

Cited by in corpus (6)

• Deviation inequalities and CLT for random walks on acylindrically hyperbolic groups
• Bounded Cohomology Classes of Exact Forms
• Borel and volume classes for dense representations of discrete groups
• Acylindrical hyperbolicity of cubical small-cancellation groups
• Median quasimorphisms on CAT(0) cube complexes and their cup products
• The space of non-extendable quasimorphisms