Algorithmic Properties of Relatively Hyperbolic Groups

TL;DR

This paper generalizes known results on biautomaticity of hyperbolic groups and their extensions, demonstrating that certain relative hyperbolic groups and their cohomological extensions preserve biautomaticity.

Contribution

It proves new theorems showing biautomaticity of extensions of hyperbolic and relatively hyperbolic groups under specific conditions.

Findings

Extensions of geometrically finite hyperbolic groups are biautomatic when certain cohomological conditions are met.

Relatively hyperbolic groups with bounded coset penetration and biautomatic peripheral subgroups are biautomatic.

Conjecture that extensions of relatively hyperbolic groups with biautomatic peripheral groups are also biautomatic.

Abstract

The following discourse is inspired by the works on hyperbolic groups of Epstein, and Neumann/Reeves. Epstein showed that geometrically finite hyperbolic groups are biautomatic. Neumann/Reeves showed that virtually central extensions of word hyperbolic groups are biautomatic. We prove the following generalisation: Theorem. Let H be a geometrically finite hyperbolic group. Let sigma in H^2(H) and suppose that sigma restricted to P is zero for any parabolic subgroup P of H. Then the extension of H by sigma is biautomatic. We also prove another generalisation of the result of Epstein. Theorem. Let G be hyperbolic relative to H, with the bounded coset penetration property. Let H be a biautomatic group with a prefix-closed normal form. Then G is biautomatic. Based on these two results, it seems reasonable to conjecture the following (which the author believes can be proven with a simple generalisation of the argument in Section 1): Let G be hyperbolic relative to H, where H has a prefixed closed biautomatic structure. Let sigma in H^2(G) and suppose that sigma restricted to H is zero. Then the extension of G by sigma is biautomatic.