An experimental investigation of Neumann's conjecture

Abstract

We use a large census of hyperbolic 3-manifolds to experimentally investigate a conjecture of Neumann regarding the Bloch Group. We present an augmented census including, for feasible invariant trace fields, explicit manifolds (associated to that field) that appear to generate the Bloch group of that field. We also make use of Ptolemy coordinates to compute "exotic volumes" of representations, and attempt to realize these volumes as linear combinations of generator volumes. We thus present a large body of empirical support for Neumann's conjecture.