Tube volumes and small hyperbolic 3-manifolds

A complete hyperbolic three-manifold M can be defined as a quotient space $H3$/G, where $H3$ is the upper half-space {lcub}(x, y, t)|t > 0{rcub} with the Riemannian metric ds2 = ( dx2 + dy2 + dt2)/t2 and G < Isom($H3$) is a discrete group which acts freely. With these restrictions on G, the quotient map from $H3$ to M is a local isometry, and hence M inherits the metric ds; the volume of M is then defined to be Vol(M, ds).; By the rigidity theorems of Mostow and Prasad, this volume is not just a geometric invariant of the pair (M, ds) but is a topological invariant of M alone. This raises the issue of determining the possible values of this invariant; in particular, one would like to know the value of V₀ = inf{lcub}Vol(M){rcub}, the minimum possible hyperbolic volume. But unlike the case in two dimensions where the problem is answered by the Gauss-Bonnet formula, the exact value of V₀ is still an open problem.; In an attempt to tackle this problem, we first demonstrate a geometric connection between the volume of a hyperbolic three-manifold and the length and tube radius of its shortest closed geodesic curve. These results are similar in nature to those we published in the 2001 paper “Volumes of tubes in hyperbolic 3-manifolds”, but are somewhat stronger; we also extend these results to non-orientable manifolds.; We then use these theoretical results as the foundation for a computer-aided search for small hyperbolic three-manifolds. In particular, we show how the theoretical results referred to above allow us to state that the fundamental group G of a small orientable hyperbolic three-manifold M must have a two-generator subgroup lying in a particular compact subset of the space of all two-generator subgroups of Isom+($H3$). We then show how to rigorously demonstrate via a computer program that large portions of this compact set cannot possibly contain any points corresponding to discrete subgroups, further narrowing the search for small hyperbolic three-manifolds.; Finally the results of this long-running computer search are presented. While the result sought was not obtained, partial results can be extracted from the data.