## Regularity of quasigeodesics in a hyperbolic group

### Derek F.\ Holt and Sarah Rees\

#### Keywords

quasigeodesics, hyoerbolic groups, automatic structures, automatic groups

#### Status

Internat. J. Algebra Comput. 13 (2003), no. 5, 585--596.

#### Abstract

We prove that for $\lambda \geq 1$ and all sufficiently large $\epsilon$, the set of \Le-quasigeodesics in an infinite word-hyperbolic group $G$ is regular if and only if $\lambda$ is rational. In fact, this set of quasigeodesics defines an asynchronous automatic structure for $G$. We also introduce the idea of an {\em exact} \Le-quasigeodesic and show that for rational $\lambda$ and appropriate $\epsilon$ the sets of exact \Le-quasigeodesics define synchronous automatic structures. \

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