The Tits alternative for generalized tetrahedron groups
James Howie, Natalia Kopteva
School of Mathematical & Computer Sciences
A generalized tetrahedron group is the colimit of a triangle of groups whose vertex groups are generalized triangle groups and whose edge groups are finite cyclic. We prove an improved spelling theorem for generalized triangle groups which enables us to compute the precise Gersten-Stallings angles of this triangle of groups, and hence obtain a classification of generalized tetrahedron groups according to the curvature properties of the triangle. We also prove that the colimit of a negatively curved triangle of groups contains a non-abelian free subgroup. Finally, we apply these results to prove the Tits alternative for all generalized tetrahedron groups where the triangle is non-spherical: with three abelian-by-finite exceptions, every such group contains a non-abelian free subgroup. © de Gruyter 2006.
Journal of Group Theory
https://doi.org/10.1515/JGT.2006.012
Published - 21 Mar 2006
10.1515/JGT.2006.012
Dive into the research topics of 'The Tits alternative for generalized tetrahedron groups'. Together they form a unique fingerprint.
Triangular pyramid Mathematics 100%
Alternatives Mathematics 53%
Triangle Mathematics 41%
Triangle Group Mathematics 26%
Colimit Mathematics 25%
Subgroup Mathematics 12%
Exception Mathematics 9%
Angle Mathematics 8%
Howie, J., & Kopteva, N. (2006). The Tits alternative for generalized tetrahedron groups. Journal of Group Theory, 9(2), 173-189. https://doi.org/10.1515/JGT.2006.012
Howie, James ; Kopteva, Natalia. / The Tits alternative for generalized tetrahedron groups. In: Journal of Group Theory. 2006 ; Vol. 9, No. 2. pp. 173-189.
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title = "The Tits alternative for generalized tetrahedron groups",
abstract = "A generalized tetrahedron group is the colimit of a triangle of groups whose vertex groups are generalized triangle groups and whose edge groups are finite cyclic. We prove an improved spelling theorem for generalized triangle groups which enables us to compute the precise Gersten-Stallings angles of this triangle of groups, and hence obtain a classification of generalized tetrahedron groups according to the curvature properties of the triangle. We also prove that the colimit of a negatively curved triangle of groups contains a non-abelian free subgroup. Finally, we apply these results to prove the Tits alternative for all generalized tetrahedron groups where the triangle is non-spherical: with three abelian-by-finite exceptions, every such group contains a non-abelian free subgroup. {\textcopyright} de Gruyter 2006.",
author = "James Howie and Natalia Kopteva",
doi = "10.1515/JGT.2006.012",
journal = "Journal of Group Theory",
Howie, J & Kopteva, N 2006, 'The Tits alternative for generalized tetrahedron groups', Journal of Group Theory, vol. 9, no. 2, pp. 173-189. https://doi.org/10.1515/JGT.2006.012
The Tits alternative for generalized tetrahedron groups. / Howie, James; Kopteva, Natalia.
In: Journal of Group Theory, Vol. 9, No. 2, 21.03.2006, p. 173-189.
T1 - The Tits alternative for generalized tetrahedron groups
AU - Howie, James
AU - Kopteva, Natalia
N2 - A generalized tetrahedron group is the colimit of a triangle of groups whose vertex groups are generalized triangle groups and whose edge groups are finite cyclic. We prove an improved spelling theorem for generalized triangle groups which enables us to compute the precise Gersten-Stallings angles of this triangle of groups, and hence obtain a classification of generalized tetrahedron groups according to the curvature properties of the triangle. We also prove that the colimit of a negatively curved triangle of groups contains a non-abelian free subgroup. Finally, we apply these results to prove the Tits alternative for all generalized tetrahedron groups where the triangle is non-spherical: with three abelian-by-finite exceptions, every such group contains a non-abelian free subgroup. © de Gruyter 2006.
AB - A generalized tetrahedron group is the colimit of a triangle of groups whose vertex groups are generalized triangle groups and whose edge groups are finite cyclic. We prove an improved spelling theorem for generalized triangle groups which enables us to compute the precise Gersten-Stallings angles of this triangle of groups, and hence obtain a classification of generalized tetrahedron groups according to the curvature properties of the triangle. We also prove that the colimit of a negatively curved triangle of groups contains a non-abelian free subgroup. Finally, we apply these results to prove the Tits alternative for all generalized tetrahedron groups where the triangle is non-spherical: with three abelian-by-finite exceptions, every such group contains a non-abelian free subgroup. © de Gruyter 2006.
U2 - 10.1515/JGT.2006.012
DO - 10.1515/JGT.2006.012
JO - Journal of Group Theory
JF - Journal of Group Theory
Howie J, Kopteva N. The Tits alternative for generalized tetrahedron groups. Journal of Group Theory. 2006 Mar 21;9(2):173-189. https://doi.org/10.1515/JGT.2006.012