Abstract
We identify the two minimal co-volume lattices of the isometry group of hyperbolic $3$-space that contain a finite spherical triangle group. These two groups are arithmetic and are in fact the two minimal co-volume lattices. Our results here represent the key step in establishing this fact, thereby solving a problem posed by Siegel in 1945. As a consequence we obtain sharp bounds on the order of the symmetry group of a hyperbolic $3$-manifold in terms of its volume, analogous to the Hurwitz $84g-84$ theorem of 1892.
The finite spherical subgroups of a Kleinian group give rise to the vertices of the singular graph in the quotient orbifold. We identify the small values of the discrete spectrum of hyperbolic distances between these vertices and show these small values give rise to arithmetic lattices. Once vertices are sufficiently separated, one obtains volume bounds by studying equivariant sets.