5 J an 1 99 4 Automatic Structures , Rational Growth , and Geometrically Finite Hyperbolic Groups

We show that the set SA(G) of equivalence classes of synchronously automatic structures on a geometrically finite hyperbolic group G is dense in the product of the sets SA(P) over all maximal parabolic subgroups P. The set BSA(G) of equivalence classes of biautomatic structures on G is isomorphic to the product of the sets BSA(P) over the cusps (conjugacy classes of maximal parabolic subgroups) of G. Each maximal parabolic P is a virtually abelian group, so SA(P) and BSA(P) were computed in [NS1]. We show that any geometrically finite hyperbolic group has a generating set for which the full language of geodesics for G is regular. Moreover, the growth function of G with respect to this generating set is rational. We also determine which automatic structures on such a group are equivalent to geodesic ones. Not all are, though all biautomatic structures are.