TRUNCATED TETRAHEDRA AND THEIR REFLECTION GROUPS

We outline the classification, up to isometry, of all tetrahedra in hy perbolic space with one or more vertices truncated, for which the dihe dral angles along the edges formed by the truncations are all pi/2, an d those remaining are all submultiples of pi. We show how to find the volumes of these polyhedra, and find presentations and small generatin g sets for the orientation-preserving subgroups of their reflection gr oups. For particular families of these groups, we find low index torsi on free subgroups, and construct associated manifolds and manifolds wi th boundary. In particular, for each g greater than or equal to 2, we find a sequence of hyperbolic manifolds with totally geodesic boundary of genus g, which we conjecture to be of least volume among such mani folds.