Algorithmic Properties of Relatively Hyperbolic Groups

The following discourse is inspired by the works on hyperbolic groups of Epstein, and Neumann/Reeves. In [E], it is shown that geometrically finite hyperbolic groups are biautomatic. In [NR1], it is shown that virtually central extensions of word hyperbolic groups are biautomatic. We prove the following generalisation: Theorem 1. Let H be a geometrically finite hyperbolic group. Let σ ∈ H(H) and suppose that σ|P = 0 for any parabolic subgroup P of H. Then the extension of H by σ is biautomatic We also prove another generalisation of the result in [E]. Theorem 2. 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 1): Let G be hyperbolic relative to H , where H has a prefixed closed biautomatic structure. σ ∈ H(G) and suppose that σ|H = 0. Then the extension of G by σ is biautomatic.