15 August 2015 The complex volume of SL(n,C)-representations of 3-manifolds
Stavros Garoufalidis, Dylan P. Thurston, Christian K. Zickert
Duke Math. J. 164(11): 2099-2160 (15 August 2015). DOI: 10.1215/00127094-3121185


For a compact 3-manifold M with arbitrary (possibly empty) boundary, we give a parameterization of the set of conjugacy classes of boundary-unipotent representations of π1(M) into SL(n,C). Our parameterization uses Ptolemy coordinates, which are inspired by coordinates on higher Teichmüller spaces due to Fock and Goncharov. We show that a boundary-unipotent representation determines an element in Neumann’s extended Bloch group Bˆ(C), and we 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.


Download Citation

Stavros Garoufalidis. Dylan P. Thurston. Christian K. Zickert. "The complex volume of SL(n,C)-representations of 3-manifolds." Duke Math. J. 164 (11) 2099 - 2160, 15 August 2015. https://doi.org/10.1215/00127094-3121185


Received: 10 September 2013; Revised: 4 September 2014; Published: 15 August 2015
First available in Project Euclid: 13 August 2015

zbMATH: 1335.57034
MathSciNet: MR3385130
Digital Object Identifier: 10.1215/00127094-3121185

Primary: 57M27 , 57M50 , 57N10 , 58J28
Secondary: 11G55 , 11R70 , 19F27

Keywords: $\operatorname{SL} (n,\mathbb{C})$-representations , algebraic $K$-theory , census manifolds , Cheeger–Chern–Simons class , Chern–Simons invariant , complex volume , extended Bloch group , hyperbolic $3$-manifolds , Ptolemy coordinates , Rogers dilogarithm , SnapPy

Rights: Copyright © 2015 Duke University Press


This article is only available to subscribers.
It is not available for individual sale.

Vol.164 • No. 11 • 15 August 2015
Back to Top