Regular languages for contracting geodesics

Abstract

Let G $G$ be a finitely generated group. We show that for any finite symmetric generating set A $A$ , the language consisting of all geodesics in Cay ( G , A ) $(G,A)$ with the contracting property is a regular language. An immediate consequence is that the existence of an infinite contracting geodesic in a Cayley graph of a finitely generated group implies the existence of a contracting element. In particular, torsion groups cannot contain an infinite contracting geodesic. As an application, this implies that any finitely generated group containing an infinite contracting geodesic must be either virtually Z $\mathbb {Z}$ or acylindrically hyperbolic.