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On free subgroups of generalized triangle groups. (English) Zbl 0694.20016
Algebra Logic 28, No. 2, 152-161 (1989); and Algebra Logika 28, No. 2, 227-240 (1989).
Let \(G=\langle a_ 1,...,a_ n\mid a_ 1^{e_ 1}=...a_ n^{e_ n}=R^ m(a_ 1,...,a_ n)=1\rangle\), 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 on \(a_ 1,...,a_ n\) involving all \(a_ 1,...,a_ n\). In this paper the author is concerned with the question: does the Tits alternative hold for \(G\) (i.e. does \(G\) contain either a free subgroup of rank 2 or a solvable subgroup of finite index)?
In an earlier joint paper with B. Fine and F. Levin [Arch. Math. 50, 97-109 (1988; Zbl 0639.20016)] the author proved, for example, that the Tits alternative holds for \(G\) provided \(n\geq 3\) or (\(n=2\) and \(m\geq 3\)). In this paper he extends this result to the case where \(n=m=2\) and \(R(a,b)=a^{\alpha}b^{\beta}a^{\gamma}b^{\delta}\), with \(1\leq\alpha,\gamma<e_ 1\) and \(1\leq \beta,\delta<e_ 2\). The proofs involve representing \(G\) as a subgroup of \(\text{PSL}_ 2(\mathbb{C})\).
Reviewer: A.W.Mason

MSC:
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20F05 Generators, relations, and presentations of groups
20E07 Subgroup theorems; subgroup growth
20H05 Unimodular groups, congruence subgroups (group-theoretic aspects)
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References:
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