Regular languages for contracting geodesics

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 ( ) $(G,A)$ with contracting property is regular language. An immediate consequence existence an infinite geodesic Cayley graph group implies element. In particular, torsion groups cannot contain geodesic. As application, this containing must either virtually Z $\mathbb {Z}$ or acylindrically hyperbolic.