The complex volume of SL(n,C)-representations of 3-manifolds

TL;DR

This paper introduces a parametrization of boundary-unipotent representations of 3-manifold groups into SL(n,C) using Ptolemy coordinates, linking them to Neumann's extended Bloch group and providing formulas for volume calculations.

Contribution

It provides a new parametrization method for boundary-unipotent representations using Ptolemy coordinates and connects these to the extended Bloch group for volume computations.

Findings

Boundary-unipotent representations are abundant in census manifolds.

The volume of a representation can be expressed as an integral linear combination of hyperbolic volumes.

Numerical evidence supports the conjecture that the Bloch group is generated by hyperbolic manifolds.

Abstract

For a compact 3-manifold M with arbitrary (possibly empty) boundary, we give a parametrization of the set of conjugacy classes of boundary-unipotent representations of the fundamental group of M into SL(n,C). Our parametrization uses Ptolemy coordinates, which are inspired by coordinates on higher Teichmueller spaces due to Fock and Goncharov. We show that a boundary-unipotent representation determines an element in Neumann's extended Bloch group, and use this to obtain an efficient formula for the Cheeger-Chern-Simons invariant, and in particular for the volume. Computations for the census manifolds show that boundary-unipotent representations are abundant, and numerical comparisons with census volumes, suggest that the volume of a representation is an integral linear combination of volumes of hyperbolic 3-manifolds. This is in agreement with a conjecture of Walter Neumann, stating that the Bloch group is generated by hyperbolic manifolds.