This note is based on J. W. Cannon's paper [2], in which he gives, among other things, a solution of the word problem for cocompact groups of isometrics of hyperbolic space of dimension d >1 1. We have thought it worthwhile to show how Cannon's ideas can be rearranged to give a rather simple solution of this problem. Although the result obtained here is in important respects weaker than that of Cannon, it is in other respects more general. In particular, it also works for cocompact groups of isometries of Euclidean space and discrete groups ofisometries of hyperbolic space whose quotients have finite volume. If one accepts as part of the initial information the Nielsen region for G, then it can also be used to give a solution of the word problem for geometrically finite groups of isometrics of hyperbolic 3-space.
We view our result as an instance of the Todd-Coxeter coset enumeration algorithm (see [5] for a good description), which we formulate as follows. Let a group G have a presentation (X:R), with G = F/N, where F is a free group with basis X and N is the normal closure of the subset R in F. For any natural number n, let R(n) be the set of all words in X of the form w = uru-1 for r in R and u an element of F represented by a reduced word of length l ul ~< n. Let F(n) be the set of all initial segments of words w in R(n). Let w ~ n w' be the equivalence relation on F(n) defined by setting wlw2 ~ nW~WoW2 whenever both are in F(n) and w 0 is in R(n).