# 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.

## Key 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.