A periodicity theorem for acylindrically hyperbolic groups

TL;DR

This paper extends a classical periodicity lemma from free groups to acylindrically hyperbolic groups, enabling new insights into solutions of equations and subgroup characterizations within these complex groups.

Contribution

It introduces a generalized periodicity theorem for acylindrically hyperbolic groups, broadening the scope of classical results to a more general class of groups.

Findings

Generalization of the periodicity lemma to acylindrically hyperbolic groups

Application to describing solutions of equations in these groups

Characterization of verbally closed finitely generated subgroups

Abstract

We generalize a well known periodicity lemma from the case of free groups to the case of acylindrically hyperbolic groups. This generalization will be used later to describe solutions of certain equations in acylindrically hyperbolic groups and to characterize verbally closed finitely generated acylindrically hyperbolic subgroups of finitely presented groups.