Some challenging group presentations

We study some challenging presentations which arise as groups of deficiency zero. In four cases we settle finiteness: we show that two presentations a re for finite groups while two are fur infinite groups. Thus we answer thre e explicit questions in the literature and we provide the first published d eficiency zero presentation for a group with derived length seven. The tool s we use are coset enumeration and Knuth-Bendix rewriting, which are well-e stablished as methods for proving finiteness or otherwise of a finitely pre sented group. We briefly comment on their capabilities and compare their pe rformance.