Automatic structures, rational growth and geometrically finite hyperbolic groups

Abstract

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.

Plain Tex, 26 pages, no figures

Cited by in corpus (32)

• Artin groups of large type are shortlex automatic with regular geodesics
• Finite generating sets of relatively hyperbolic groups and applications to geodesic languages
• On groups whose geodesic growth is polynomial
• Some geodesic problems in groups
• Regular geodesic languages and the falsification by fellow traveler property
• Finiteness and the falsification by fellow traveler property
• Counting elements and geodesics in Thompson's group $F$
• The loop shortening property and almost convexity
• Cone types and geodesic languages for lamplighter groups and Thompson's group F
• The rationality of Sol manifolds
• Rational growth in the Heisenberg group
• Injective hulls of certain discrete metric spaces and groups
• Regularity of quasigeodesics characterises hyperbolicity
• Graphs with convex balls
• Counting subgraphs in fftp graphs with symmetry
• Patterns theory and geodesic automatic structure for a class of groups
• A non-Hopfian almost convex group
• Locally elliptic actions, torsion groups, and nonpositively curved spaces
• Conjugacy languages in groups
• Conjugacy languages in virtual graph products
• Rational cross-sections, bounded generation and orders on groups
• Quantifying the Residual Properties of Gamma-Limit Groups
• A virtually 2-step nilpotent group with polynomial geodesic growth
• Pascal's Triangles in Abelian and Hyperbolic Groups
• Triangles, squares and geodesics
• Conjugacy growth series and languages in groups
• Geodesic growth in right-angled and even Coxeter groups
• Rational Growth and Almost Convexity of Higher-Dimensional Torus Bundles
• Piecewise excluding geodesic languages
• Groups whose geodesics are locally testable
• Star-free geodesic languages for groups
• On a generalization of Dehn's algorithm