Logged rewriting and identities among relators

Summary

Abstract

We present a version of the Knuth-Bendix string rewriting procedures for group computations and apply it to the problem of computing the module of identities among relators. By lifting rewriting into the appropriate higher dimension we provide a methodology which is alternative and complementary to the popular geometric approach of pictures.

Introduction

Combinatorial group theory is the study of groups which are given by means of presentations; these arise naturally in a wide variety of situations including areas as diverse as knot theory [13], geometry [8] and cryptography [1]. One of the fundamental problems in computational group theory is the solution of the word problem for a given presentation. The problem is in general undecidable and consequently a number of different approaches have been developed. Amongst the most successful is string rewriting, in particular Knuth-Bendix completion, which attempts to solve the word problem by trying to generate a confluent and Noetherian rewrite system from the presentation. The advantages of this approach are twofold: i) Knuth-Bendix completion can be successfully applied in a large number of situations and; ii) the concrete nature of string rewriting makes these algorithms relatively easy to implement. Indeed, many computer algebra packages solve word problems in precisely this way [11, 18].

Every presentation has associated with it a CW-complex: a cellular model whose fundamental group is the group given by the presentation. The second homotopy group of the CW-complex is the module of identities among relators.