Regular Languages for Contracting Geodesics

TL;DR

This paper proves that the set of contracting geodesics in a finitely generated group forms a regular language and explores implications for the group's structure, such as being virtually  or acylindrically hyperbolic.

Contribution

It establishes the regularity of contracting geodesic languages and characterizes groups with infinite contracting geodesics.

Findings

Contracting geodesic languages are regular.

Groups with infinite contracting geodesics are either virtually  or acylindrically hyperbolic.

Provides a link between geometric properties and algebraic group classifications.

Abstract

Let $G$ be a finitely generated group. We show that for any finite generating set $A$, the language consisting of all geodesics in $Cay(G,A)$ with a contracting property is a regular language. As an application, we show that any finitely generated group containing an infinite contracting geodesic must be either virtually $\mathbb{Z}$ or acylindrically hyperbolic.