Groups -- Complexity -- Cryptology 01 (2009), No. 2, 207--216
Copyright Heldermann Verlag 2009
The Tits Alternative for Tsaranov's Generalized Tetrahedron Groups
Volkmar Große Rebel
Fakultät für Mathematik, Technische Universität, 44221 Dortmund, Germany
volkmar.rebel@tecis.de
Miriam Hahn
Fakultät für Mathematik, Technische Universität, 44221 Dortmund, Germany
miriam.hahn@mathematik.uni-dortmund.de
Gerhard Rosenberger
Fakultät für Mathematik, Technische Universität, 44221 Dortmund, Germany
rosenberger@mathematik.uni-dortmund.de
[Abstract-pdf]
A generalized tetrahedron group is defined to be a group admitting the following presentation: $\langle x,y,z\mid x^l=y^m= z^n=W_1^p(x,y)=W_2^q(y,z)=W_3^r(x,z)=1\rangle$, $2\le l,m,n,p,q,r$, 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.\par In this paper, we build on previous work to show that the Tits alternative holds for Tsaranov's generalized tetrahedron groups, that is, if $G$ is a Tsaranov generalized tetrahedron group then $G$ contains a non-abelian free subgroup or is solvable-by-finite. The term {\em Tits alternative} comes from the respective property for finitely generated linear groups over a field (see J. Tits [Free subgroups in linear groups, J. Algebra 20 (1972) 250--270]).
Keywords: Generalized tetrahedron groups, generalized triangle groups, the Tits alternative, triangle of groups, Fortsetzungssatz.
MSC: 20F05; 20E05, 20E07
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