  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 ``Equivalent
automatic structures and their boundaries'' by M.Shapiro and W.Neumann, Intern.
J. of Alg. Comp. 2 (1992) 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.
