The Tits alternative for short generalized tetrahedron groups.
A well-known theorem of \textit{J. Tits} [J. Algebra 20, 250-270 (1972; Zbl 0236.20032)] states that a finitely generated linear group over a field either is soluble-by-finite or contains a non-cyclic free subgroup. In this paper the authors prove that this Tits alternative also holds for short generalized tetrahedron groups. Here a group \(G\) defined by a presentation \(G=\langle x,y,z\mid x^l=y^m=z^n=W_1^p(x,y)=W_2^q(y,z)=W_3^r(x,z)=1\rangle\), \(2\leq l,m,n,p,q,r\), where each \(W_i(a,b)\) is a cyclically reduced word involving both \(a\) and \(b\), is called a generalized tetrahedron group. The group \(G\) is a short generalized tetrahedron group if all the \(W_i(a,b)\) have length at most \(4\) in the respective free product on \(a,b\).
scientific article; zbMATH DE number 6076257