Duke Mathematical Journal

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

Stavros Garoufalidis, Dylan P. Thurston, and Christian K. Zickert

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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.

Article information

Duke Math. J., Volume 164, Number 11 (2015), 2099-2160.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57N10: Topology of general 3-manifolds [See also 57Mxx] 57M27: Invariants of knots and 3-manifolds 57M50: Geometric structures on low-dimensional manifolds 58J28: Eta-invariants, Chern-Simons invariants
Secondary: 11R70: $K$-theory of global fields [See also 19Fxx] 19F27: Étale cohomology, higher regulators, zeta and L-functions [See also 11G40, 11R42, 11S40, 14F20, 14G10] 11G55: Polylogarithms and relations with $K$-theory

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


Garoufalidis, Stavros; Thurston, Dylan P.; Zickert, Christian K. The complex volume of $\operatorname {SL}(n,\mathbb{C})$ -representations of 3-manifolds. Duke Math. J. 164 (2015), no. 11, 2099--2160. doi:10.1215/00127094-3121185. https://projecteuclid.org/euclid.dmj/1439470580

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