Algorithmic Properties of Relatively Hyperbolic Groups

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.