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

By Stavros Garoufalidis, Dylan Thurston, C. Zickert. 2011

75 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e818f49a63ab7334bc64ad8bd4d3512554ef9a43

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Citations in the corpus

3-Manifolds and 3d Indices (2011, background): “uch theories would arise from a compacti- cation of KM5 branes on a three-manifold M. Mathematically, some extensions of hyperbolic invariants of 3-manifolds to higher rank have been investigated in [39,40], following [41], and are known to still lie in the classical Bloch group (cf. Footnote2). The”
Gluing equations for PGL(n, ℂ)–representations of 3–manifolds (2012, methodology): “In [12] we parametrized boundary-unipotent representations of a 3-manifold group into SL(n,C) using Ptolemy coordinates, which were inspired by A-coordinates on higher Teichmüller space due to Fock and Goncharov.”
On the asymptotic expansion of the Kashaev invariant of the $5_2$ knot (2016, background): “[10], that the complex volume can be regarded as the SL(2,C) Chern-Simons invariant.”