Regularity of quasigeodesics characterises hyperbolicity

TL;DR

This paper characterizes hyperbolic groups through the regularity of quasigeodesics in Cayley graphs and provides a quantitative measure of hyperbolicity based on local quasigeodesic loops, advancing understanding of group geometry.

Contribution

It introduces a new characterization of hyperbolic groups using regular languages of quasigeodesics and offers a quantitative approach to hyperbolicity in metric spaces.

Findings

Hyperbolic groups correspond to regular languages of quasigeodesics.

A quantitative criterion for hyperbolicity based on local quasigeodesic loops.

Progress on Shapiro's question about uniquely geodesic Cayley graphs.

Abstract

We characterise hyperbolic groups in terms of quasigeodesics in the Cayley graph forming regular languages. We also obtain a quantitative characterisation of hyperbolicity of geodesic metric spaces by the non-existence of certain local (3,0)-quasigeodesic loops. As an application we make progress towards a question of Shapiro regarding groups admitting a uniquely geodesic Cayley graph.