# J. Howie, Generalised Tetrahedron Groups and the Tits Alternative

There are standard presentations for the group if isometries
of hyperbolic, euclidean or spherical space generated by reflections
in the faces of a tetrahedron whose dihedral angles are submultiples
of $\pi$, and for its index $2$ subgroup of orientation-preserving isometries.
A generalised tetrahedron group is one with a somewhat more general
presentation. These are colimits of triangles of generalised triangle
groups. I shall explain how to calculate precisely the Gersten-Stallings
angles of this triangle, and why when the triangle is non-spherical
the group contains a non-abelian free subgroup (with three exceptions,
where the group is virtually abelian). If time permits, I shall also
present evidence for the Tits alternative for this class of groups
even where the triangle is spherical.

A.J. Duncan
Last modified: Wed Jun 2 17:55:53 BST 2004