An experimental investigation of Neumann's conjecture

TL;DR

This paper provides extensive empirical evidence supporting Neumann's conjecture by analyzing hyperbolic 3-manifolds, computing exotic volumes, and identifying generators of Bloch groups across various invariant trace fields.

Contribution

It introduces an augmented census of hyperbolic 3-manifolds with explicit generators and employs Ptolemy coordinates to compute exotic volumes, offering new empirical support for Neumann's conjecture.

Findings

Identification of explicit manifolds generating Bloch groups

Computation of exotic volumes of representations

Empirical support for 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.