Algebraic & Geometric Topology

Gluing equations for $\mathrm{PGL}(n,\mathbb{C})$–representations of $3$–manifolds

Abstract

Garoufalidis, Thurston and Zickert 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. We parametrize representations into $PGL(n, ℂ)$ using shape coordinates, which are a $3$–dimensional analogue of Fock and Goncharov’s $X$–coordinates. These coordinates satisfy equations generalizing Thurston’s gluing equations. These equations are of Neumann–Zagier type and satisfy symplectic relations with applications in quantum topology. We also explore a duality between the Ptolemy coordinates and the shape coordinates.

Article information

Source
Algebr. Geom. Topol., Volume 15, Number 1 (2015), 565-622.

Dates
Received: 7 November 2014
Accepted: 13 December 2014
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1510840923

Digital Object Identifier
doi:10.2140/agt.2015.15.565

Mathematical Reviews number (MathSciNet)
MR3325748

Zentralblatt MATH identifier
1347.57014

Citation

Garoufalidis, Stavros; Goerner, Matthias; Zickert, Christian. Gluing equations for $\mathrm{PGL}(n,\mathbb{C})$–representations of $3$–manifolds. Algebr. Geom. Topol. 15 (2015), no. 1, 565--622. doi:10.2140/agt.2015.15.565. https://projecteuclid.org/euclid.agt/1510840923

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