Regularity of Quasigeodesics in a Hyperbolic Group

We prove that for λ≥1 and all sufficiently large ∊, the set of (λ,∊)-quasigeodesics in an infinite word-hyperbolic group G is regular if and only if λ is rational. In fact, this set of quasigeodesics defines an asynchronous automatic structure for G. We also introduce the idea of an exact (λ,∊)-quasigeodesic and show that for rational λ and appropriate ∊ the sets of exact (λ,∊)-quasigeodesics define synchronous automatic structures.