Cogrowth for group actions with strongly contracting elements

TL;DR

This paper establishes a lower bound on the growth rate ratio of an infinite normal subgroup to the whole group for groups acting with strongly contracting elements, extending classical results to a broader class of groups and spaces.

Contribution

It proves that for groups with purely exponential growth acting with strongly contracting elements, the subgroup's growth rate exceeds half of the group's growth rate, generalizing prior results.

Findings

If G has purely exponential growth, then δ_N/δ_G > 1/2.

Applicable to hyperbolic, CAT(0), and other groups with suitable actions.

Extends classical growth rate results to wider group actions.

Abstract

Let $G$ be a group acting properly by isometries and with a strongly contracting element on a geodesic metric space. Let $N$ be an infinite normal subgroup of $G$, and let $\delta_N$ and $\delta_G$ be the growth rates of $N$ and $G$ with respect to the pseudo-metric induced by the action. We prove that if $G$ has purely exponential growth with respect to the pseudo-metric then $\delta_N/\delta_G>1/2$. Our result applies to suitable actions of hyperbolic groups, right-angled Artin groups and other CAT(0) groups, mapping class groups, snowflake groups, small cancellation groups, etc. This extends Grigorchuk's original result on free groups with respect to a word metrics and a recent result of Jaerisch, Matsuzaki, and Yabuki on groups acting on hyperbolic spaces to a much wider class of groups acting on spaces that are not necessarily hyperbolic.