The Tits Alternative for Groups Defined by Periodic Paired Relations

ABSTRACT

The class of groups defined by periodic paired relations, as introduced by Vinberg, includes the generalized triangle groups, the generalized tetrahedron groups, and the generalized Coxeter groups. We observe that any group defined by periodic paired relations Γ can be realized as a so-called “Pride group”. Using results of Howie and Kopteva we give necessary and sufficient conditions for this Pride group to be non-spherical. Under such conditions, we show that Γ satisfies the Tits alternative.

Communicated by A. Olshanskiy

Keywords:

• Generalized Coxeter group
• Generalized tetrahedron group
• Generalized triangle group
• Group defined by periodic paired relations
• Pride group
• Tits alternative

Mathematics Subject Classification:

• Primary 20F05
• Secondary 20F55
• 20F65
• 20H15

ACKNOWLEDGMENTS

I would like to thank Jim Howie and Natasha Kopteva for helpful discussions in connection with this work. This work was carried out at the National University of Ireland, Galway, while supported by Marie Curie fellowship HPMD-CT- 2001-00079.