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**The complex volume of \(\mathrm{SL}(n,\mathbb{C})\)-representations of 3-manifolds.**
*(English)*
Zbl 1335.57034

Representations of \(3\)–manifold groups into \(SL(2,\mathbb{C})\) have been intensively studied for both classical reasons (e.g. CR geometry) and quantum reasons (e.g. the Volume Conjecture). Representations of surface groups into algebraic groups have also been intensively studied.

This paper under review extends Ptolemy coordinates, first introduced in the \(SL(2,\mathbb{C})\) context, to the \(SL(n,\mathbb{C})\) case. These are are \(3\)–dimensional analogues of coordinates of Fock and Goncharov on higher Teichmüller spaces which parameterize the set of conjugacy classes of boundary-unipotent representations of the fundamental group of a compact \(3\)-manifold \(M\) into \(SL(n,\mathbb{C})\); Ptolemy coordinates define a variety called the ‘Ptolemy variety’ which is an invariant of a topological ideal triangulation of \(M\). This variety in turn defines an element in the extended Bloch group which in turn gives the Cheeger–Chern–Simons invariant of \(M\), echoing the \(SL(2,\mathbb{C})\) construction of W. D. Neumann [Geom. Topol. 8, 413–474 (2004; Zbl 1053.57010)].

The computational advantage of Ptolemy coordinates is that they homogenize gluing equations and lower their degree to 2, at the price of increasing the number of variables. M. Goerner computed the Ptolemy variety and its decomposition by running the algorithm in this paper on MAGMA for all census manifolds up to \(8\) simplices and for many link complements for \(n=2\), for fewer for \(n=3\), and for triangulations with two simplices for \(n=4\). These appear to be the first computations of Cheeger–Chern–Simons invariants for \(n>2\). These computations support a conjecture of Walter Neumann stating that the Bloch group is generated by Bloch invariants of hyperbolic manifolds.

This paper under review extends Ptolemy coordinates, first introduced in the \(SL(2,\mathbb{C})\) context, to the \(SL(n,\mathbb{C})\) case. These are are \(3\)–dimensional analogues of coordinates of Fock and Goncharov on higher Teichmüller spaces which parameterize the set of conjugacy classes of boundary-unipotent representations of the fundamental group of a compact \(3\)-manifold \(M\) into \(SL(n,\mathbb{C})\); Ptolemy coordinates define a variety called the ‘Ptolemy variety’ which is an invariant of a topological ideal triangulation of \(M\). This variety in turn defines an element in the extended Bloch group which in turn gives the Cheeger–Chern–Simons invariant of \(M\), echoing the \(SL(2,\mathbb{C})\) construction of W. D. Neumann [Geom. Topol. 8, 413–474 (2004; Zbl 1053.57010)].

The computational advantage of Ptolemy coordinates is that they homogenize gluing equations and lower their degree to 2, at the price of increasing the number of variables. M. Goerner computed the Ptolemy variety and its decomposition by running the algorithm in this paper on MAGMA for all census manifolds up to \(8\) simplices and for many link complements for \(n=2\), for fewer for \(n=3\), and for triangulations with two simplices for \(n=4\). These appear to be the first computations of Cheeger–Chern–Simons invariants for \(n>2\). These computations support a conjecture of Walter Neumann stating that the Bloch group is generated by Bloch invariants of hyperbolic manifolds.

Reviewer: Daniel Moskovich (Beer-Sheva)

### MSC:

57N10 | Topology of general \(3\)-manifolds (MSC2010) |

57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |

57M50 | General geometric structures on low-dimensional manifolds |

58J28 | Eta-invariants, Chern-Simons invariants |

11R70 | \(K\)-theory of global fields |

19F27 | Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects) |

11G55 | Polylogarithms and relations with \(K\)-theory |