Regularity of quasigeodesics characterises hyperbolicity

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.

v3: Updated with referee's comments. To appear in Proceedings of the Royal Society of Edinburgh Section A: Mathematics. v2: 14 pages, comments welcome!

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