Minimal co-volume hyperbolic lattices, II: Simple torsion in a Kleinian group

Abstract

This paper represents the final step in solving the problem, posed by Siegel in 1945, of determining the minimal co-volume lattices of hyperbolic $3$-space $\mathbb{H}$ (also Problem 3.60 (F) in the Kirby problem list from 1993). Here we identify the two smallest co-volume lattices. Both these groups are two-generator arithmetic lattices, generated by two elements of finite orders 2 and 3. Their co-volumes are $0.0390\dots$ and $ 0.0408\dots$; the precise values are given in terms of the Dedekind zeta function of a number field via a formula of Borel.
Our earlier work dealt with the cases when there is a finite spherical subgroup or high order torsion in the lattice. Thus, here we are concerned with the study of simple torsion of low order and the geometric structure of Klein 4-subgroups of a Kleinian group. We also identify certain universal geometric constraints imposed by discreteness on Kleinian groups which are of independent interest.
To obtain these results we use a range of geometric and arithmetic criteria to obtain information on the structure of the singular set of the associated orbifold and then co-volume bounds by studying equivariant neighbourhoods of fixed point sets, together with a rigorous computer search of certain parameter spaces for two-generator Kleinian groups.