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The Tits Alternative for Tsaranov's Generalized Tetrahedron Groups

The Tits Alternative for Tsaranov's Generalized Tetrahedron Groups A generalized tetrahedron group is defined to be a group admitting the following presentation: , 2 ≤ 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 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 Tits alternative comes from the respective property for finitely generated linear groups over a field (see Tits, J. Algebra 20: 250–270, 1972). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Groups - Complexity - Cryptology de Gruyter

The Tits Alternative for Tsaranov's Generalized Tetrahedron Groups

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Publisher
de Gruyter
Copyright
© Heldermann Verlag
ISSN
1867-1144
eISSN
1869-6104
DOI
10.1515/GCC.2009.207
Publisher site
See Article on Publisher Site

Abstract

A generalized tetrahedron group is defined to be a group admitting the following presentation: , 2 ≤ 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 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 Tits alternative comes from the respective property for finitely generated linear groups over a field (see Tits, J. Algebra 20: 250–270, 1972).

Journal

Groups - Complexity - Cryptology – de Gruyter

Published: Oct 1, 2009

Keywords: Generalized tetrahedron groups; generalized triangle groups; the Tits alternative; triangle of groups; Fortsetzungssatz

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