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

Stavros Garoufalidis, Matthias Goerner and Christian K Zickert

Algebraic & Geometric Topology 15 (2015) 565–622

Garoufalidis, Thurston and Zickert parametrized boundary-unipotent representations of a 3–manifold group into SL(n, ) using Ptolemy coordinates, which were inspired by A–coordinates on higher Teichmüller space due to Fock and Goncharov. We parametrize representations into PGL(n, ) 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.

generalized gluing equations, shape coordinates, Ptolemy coordinates, Neumann–Zagier datum
Mathematical Subject Classification 2010
Primary: 57M27, 57N10
Secondary: 53D50
Received: 7 November 2014
Accepted: 13 December 2014
Published: 23 March 2015
Stavros Garoufalidis
School of Mathematics
Georgia Institute of Technology
686 Cherry Street
Atlanta, GA 30332-0160
Matthias Goerner
Pixar Animation Studios
1200 Park Avenue
Emeryville, CA 94608
Christian K Zickert
Department of Mathematics
University of Maryland
College Park, MD 20742-4015