The Tits alternative for short generalized tetrahedron groups

Abstract

A generalized tetrahedron group is defined to be a group admitting the following presentation:

\(\langle x, y, z\) \(\mid x^l = y^m = z^n = W_1^p(x, y) = W_2^q(y, z) = W_3^r(x, z) = 1\) \(\rangle, 2 \leq l, m, n, p, q, r\),

where each \(W_i(a, b)\) is a cyclically reduced word involving both \(a\) and \(b\). These groups appear in many contexts, not least as fundamental groups of certain hyperbolic orbifolds or as subgroups of generalized triangle groups. In this paper, we build on previous work to show that the Tits alternative holds for short generalized tetrahedron groups, that is, if \(G\)  is a short generalized tetrahedron group then \(G\) contains a non-abelian free subgroup or is solvable-by-finite. The term Tits alternative comes from the respective property for finitely generated linear groups over a field (see [Ti]).