Confined subgroups in groups with contracting elements

TL;DR

This paper investigates the growth behavior of confined subgroups in groups with contracting elements, revealing that their growth exceeds half of the group's overall growth rate and exploring boundary action decompositions.

Contribution

It introduces new results on the growth of confined subgroups, boundary actions, and Hopf decomposition, connecting these concepts in the context of hyperbolic groups.

Findings

Confined subgroups grow faster than half the ambient group's growth rate.

Confined subgroups are shown to be conservative in boundary actions.

A dichotomy on quotient growth of Schreier graphs is established.

Abstract

In this paper, we study the growth of confined subgroups through boundary actions of groups with contracting elements. We establish that the growth rate of a confined subgroup is strictly greater than half of the ambient growth rate in groups with purely exponential growth. Along the way, several results are obtained on the Hopf decomposition for boundary actions of subgroups with respect to conformal measures. In particular, we prove that confined subgroups are conservative, and examples of subgroups with nontrivial Hopf decomposition are constructed. We show a connection between Hopf decomposition and quotient growth and provide a dichotomy on quotient growth of Schreier graphs for subgroups in hyperbolic groups.