# 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