Boundary dynamics, triple transitivity, and mixed identities in weakly hyperbolic groups

TL;DR

This paper explores the algebraic and dynamical characteristics of certain hyperbolic groups, introducing a new classification based on boundary action dynamics and establishing related algebraic properties and rigidity results.

Contribution

It introduces the lim-free criterion for classifying groups and links it to the algebraic property of being mixed identity free, extending previous results.

Findings

Satisfying the lim-free criterion is equivalent to being mixed identity free.

For non lim-free groups, all 3-transitive faithful actions are rigid and have transitivity degree at most 3.

The work generalizes prior results on hyperbolic group actions and identities.

Abstract

We study the interplay between the algebraic and dynamical properties of groups that admit a general type action on a $\delta$-hyperbolic space such that the induced action on the limit set of the Gromov boundary is faithful. We divide the class of such groups into two subclasses based on a dynamical criterion, which we call lim-free. We prove that satisfying the criterion is equivalent to a purely algebraic property of being mixed identity free, generalizing results from \cite{FMMS} and \cite{BM}. For the subclass of not lim-free groups, we give the rigidity result for all $3$-transitive faithful actions and bound the transitivity degree by $3$, generalizing the result from \cite{BM}.