A note on the Lonergan-Hosack presentation (Q1185406)

fenote by \({\mathcal G}(m)\) the group with two generators, say \(x\) and \(y\), and the defining relations \(x^ 3\cdot y^{m-4}\cdot x^ 3\cdot y^{- 1}=x^ 5\cdot y^{m-3}\cdot x^ 2\cdot y^{m-3}=e\) [This notation differs slightly from that of the authors.] The authors show that \({\mathcal G}(2)\) has order 72 and \({\mathcal G}(4)\) has order 19, and all other \({\mathcal G}(m)\) are infinite [this had been known for a few values of \(m\)]; and they show rather more, for example that they have nonabelian free subgroups and are \(SQ\)-universal.