Regularity of quasigeodesics characterises hyperbolicity
Group Theory
2025-04-14 v3
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.
Cite
@article{arxiv.2205.08573,
title = {Regularity of quasigeodesics characterises hyperbolicity},
author = {Sam Hughes and Patrick S. Nairne and Davide Spriano},
journal= {arXiv preprint arXiv:2205.08573},
year = {2025}
}
Comments
v3: Updated with referee's comments. To appear in Proceedings of the Royal Society of Edinburgh Section A: Mathematics. v2: 14 pages, comments welcome!