Generalized triangle groups of type \((2,m,2)\) (Q2702059)

A generalized triangle group is a group with a presentation \(G=\langle a,b\mid a^l=b^m=w^n=1\rangle\) where \(l,m,n\geq 2\) and \(w=a^{\alpha_1}b^{\beta_1}\cdots a^{\alpha_k}b^{\beta_k}\) with \(k\geq 1\), \(0<\alpha_i<l\), \(0<\beta_i<m\) for all \(i\). It is conjectured that \(G\) satisfies the Tits alternative, that is, \(G\) either contains a soluble subgroup of finite index or a non-Abelian free subgroup. In fact \(G\) satisfies the Tits alternative if \(k\leq 4\) or \(n\geq 3\) or \({1\over l}+{1\over m}+{1\over n}\leq 1\) [see for instance the book ``Algebraic generalizations of discrete groups: a path to combinatorial group theory through one-relator products'' by \textit{B. Fine} and \textit{G. Rosenberger}, Marcel Dekker (1999; Zbl 0933.20001)]. Here, the author concentrates essentially to the case \(l=2\) and \(m\geq 7\). If here \(k\) is even then \(G\) contains a non-Abelian free subgroup, and if here \(k\geq 3\) is odd then \(G\) has a subgroup whose pro-\(p\) completion contains a non-Abelian free subgroup for some prime \(p\).NEWLINENEWLINEFor the entire collection see [Zbl 0940.00028].