Large quotients of group actions with a contracting element

TL;DR

This paper demonstrates that groups with contracting elements acting properly on metric spaces have quotient groups whose growth rates approach the original group's growth rate, with applications to CAT(0) and mapping class groups.

Contribution

It establishes a connection between contracting elements and the approximation of a group's growth rate via quotient groups, extending previous theories with new constructions.

Findings

Existence of quotient groups with growth rates approaching the original group.

Application of the theory to CAT(0) groups and mapping class groups.

Use of advanced tools like the extension lemma and rotating families.

Abstract

For any proper action of a non-elementary group $G$ on a proper geodesic metric space, we show that if $G$ contains a contracting element, then there exists a sequence of proper quotient groups whose growth rate tends to the growth rate of $G$. Similar statements are obtained for a product of proper actions with contracting elements. The tools involved in this paper include the extension lemma for the construction of large tree, the theory of rotating families developed by F. Dahmani, V. Guirardel and D. Osin, and the construction of a quasi-tree of metric spaces introduced by M. Bestvina, K. Bromberg and K. Fujiwara. Several applications are given to CAT(0) groups and mapping class groups.