Parallel computation over hyperbolic groups

Abstract

Hyperbolic groups are a rich class of groups frequently encountered in mathematical research, particularly in topology. It has been the focus of intense study by many combinatorial group theorists and topologists recently. We present some computational results for infinite groups, especially for hyperbolic groups. It is shown that the word problem for hyperbolic groups is solvable in NC2. This is the first NC algorithm for a class of groups in combinatorial group theory. We also consider the isomorphism problem of randomly generated groups using a novel technique: the Alexander polynomial from knot theory. These randomly generated groups are almost always hyperbolic groups.