Some Challenging Group Presentations

We study some challenging presentations which arise as groups of deeciency zero. In four cases we settle niteness: we show that two presentations are for nite groups while two are for innnite groups. Thus we answer three explicit questions in the literature and we provide the rst published deeciency zero presentation for a group with soluble length seven. The tools we use are coset enumeration and Knuth-Bendix rewriting, which are well-established as methods for proving niteness or otherwise of a nitely presented group. We brieey comment on their capabilities and compare their performance.