Truncated Tetrahedra and their Reflection Groups



We classify, up to isometry, all tetrahedra in hyperbolic space with one
or more vertices truncated, for which the dihedral angles along the edges
formed by the truncations are all $pi/2$, and those remaining
are all submultiples of $pi$. We show how to find the volumes
of these polyhedra, and find presentations and small generating sets
for the orientation-preserving subgroups of their reflection groups.

For particular families of these groups, we find low index torsion free
subgroups, and construct associated manifolds and manifolds with
boundary. In particular, for each $g geq 2$, we find a sequence of
hyperbolic manifolds with totally geodesic boundary of genus $g$, which
we conjecture to be of least volume among such manifolds.

Coxter polytope, Truncated tetrahedron, Co-volume, Torsion-free subgroup, Hyperbolic Manifold

Math Review Classification
Primary 30F40, 57N10 ; Secondary 20F55

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47 pages

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