Properties of acylindrically hyperbolic groups and their small cancellation quotients

Abstract

We investigate the class of acylindrically hyperbolic groups, which includes many examples of groups which admit natural actions on hyperbolic metric spaces, such as hyperbolic and relatively hyperbolic groups, mapping class groups, and outer automorphism groups of free groups. First, we prove an extension theorem for quasi-cocyles which has applications to bounded cohomology and stable commutator length of subgroups in acylindrically hyperbolic groups. Next, we show that a version of small cancellation theory developed for hyperbolic groups and relatively hyperbolic groups by Olshankii and Osin respectively can be extended to the class of acylindrically hyperbolic groups. We give several applications of this small cancellation theory, including showing how it can be used to build various ``exotic" quotient groups. In addition, we show that these small cancellation techniques can be used to completely classify conjugacy growth functions of finitely generated groups.