Free subgroups and decompositions of one-relator products of cyclics. I: The Tits alternative (Q1100277)

Let \(G=<a_ 1,...,a_ n\); \(a_ i^{e_ i}=1\) (1\(\leq i\leq n)\), R \(m(a_ 1,...,a_ n)=1>\), where m,n\(\geq 2\), \(e_ i=0\) or \(e_ i\geq 2\) (1\(\leq i\leq n)\), and \(R(a_ 1,...,a_ n)\) is a cyclically reduced word in the free product in \(a_ 1,...,a_ n\) which involves all \(a_ 1,...,a_ n\). The authors prove that the Tits alternative holds for G (i.e. G has either a free subgroup of rank 2 or a solvable subgroup of finite index) in the following cases: (i) \(n\geq 3\), (ii) \(n=2\) and \((e_ 1=0\) or \(m\geq 2)\). The proofs involve representing G in \(PSL_ 2({\mathbb{C}})\), based on an earlier method of Baumslag, Morgan and Shalen.