Geometric Structures on 3-manifolds

A theorem of Borel's asserts that for any positive real number $V$, there are at most finitely many arithmetic lattices in ${\rm PSL}_2({\mathbb C})$ of covolume at most $V$, or equivalently at most finitely many arithmetic hyperbolid $3$-orbifolds of volume at most $V$. Determining all of these for a given $V$ is algorithmically possible for a given $V$ thanks to work by Chinburg and Friedman, but appears to be impractical except for very small values of $V$, say $V=0.41$. (The smallest covolume of a hyperbolic $3$-orbifold is about $0.39$.) It turns out that the difficulty in the computation for a larger value of $V$ can be dealt with if one can find a good bound on $\dim H_1(O,\mathbb Z/2 \mathbb Z)$, where $O$ is a hyperbolic $3$-orbifold of volume at most $V$. In the case of a hyperbolic $3$-manifold $M$, not necessarily arithmetic, joint work of mine with Marc Culler and others gives good bounds on the dimension of $H_1(M,\mathbb Z/2 \mathbb Z)$ in the presence of a suitable bound on the volume of $M$. In this talk I will discuss some analogous results for hyperbolic $3$-orbifolds, and the prospects for applying results of this kind to the enumeration of arithmetic lattices. A feature of the work that I find intriguing is that while it builds on my geometric work with Culler, the new ingredients involve primarily purely topological arguments about manifolds---the underlying spaces of the orbifolds in question---and have a classical, combinatorial flavor. At this point it appears that I can prove the following statement: If $\Omega$ is a hyperbolic 3-orbifold of volume at most $1.72$, having a link as singular set and containing no embedded turnovers, then \[\dim H_1(\Omega;\mathbb Z_2)\le 3 + 6 \bigg(\bigg\lfloor\frac{10}{3}\mathop{\rm vol}(\Omega)\bigg\rfloor+\bigg\lfloor\frac{5}{3}\mathop{\rm vol}(\Omega)\bigg\rfloor\bigg).\] In particular, $\dim H_1(\Omega;\mathbb Z_2)\le 45$. Various stronger bounds on $\dim H_1(\Omega;\mathbb Z_2)$ follow from stronger bounds on the volume of $\Omega$. The restriction on turnovers is not an obstruction to applying the results to the enumeration of arithmetic groups. The assumption that the singular set is a link is more serious, but as it is used only in a mild way in this work, the methods seem promising for the prospective application.