Genericity of contracting geodesics in groups

TL;DR

This paper characterizes the hyperbolic nature of finitely generated groups by examining the prevalence of contracting geodesics within their Cayley graphs, linking geometric properties to algebraic structure.

Contribution

It provides a new characterization of Gromov hyperbolicity based on the genericity of contracting geodesics in Cayley graphs of finitely generated groups.

Findings

Contracting geodesics are generic in hyperbolic groups.

Non-hyperbolic groups lack the genericity of contracting geodesics.

The paper establishes a criterion for hyperbolicity via geodesic behavior.

Abstract

Let G be a finitely generated group and Cay(G, S) be the Cayley graph of G with respect to a finite generating set S. We characterize the Gromov hyperbolicity of G in terms of the genericity of contracting elements in Cay(G, S).