On a Group Presentation Due to Fox

Abstract

In 1956 R. H. Fox had occasion, while investigating fundamental groups of topological surfaces, to believe that the group <a, b | ab2=b3a, ba2=a2b> was trivial. Using the Todd-Coxeter coset enumeration algorithm a proof was obtained, see [3], and this algorithmic proof was used to produce an algebraic proof, see [2]. In [1] Benson and Mendelsohn, using a similar method to that of [2] showed that <a, b | abn=bn+1a, ban=an+1b> is trivial. In this note we give a direct proof for the more general problem of describing the structure of the group <a, b | abn=bℓa, ban=aℓb>.