Geodesic
Gluing equations for PGL(n, ℂ)–representations of 3–manifolds
Garoufalidis, Stavros1 ; Goerner, Matthias2 ; Zickert, Christian3
1School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA 30332-0160, USA
2Pixar Animation Studios, 1200 Park Avenue, Emeryville, CA 94608, USA
3Department of Mathematics, University of Maryland, College Park, MD 20742-4015, USA
Algebraic and Geometric Topology, Tome 15 (2015) no. 1, pp. 565-622
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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.

DOI : 10.2140/agt.2015.15.565
Classification : 57M27, 57N10, 53D50
Keywords: generalized gluing equations, shape coordinates, Ptolemy coordinates, Neumann–Zagier datum

Garoufalidis, Stavros  1   ; Goerner, Matthias  2   ; Zickert, Christian  3

1 School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA 30332-0160, USA
2 Pixar Animation Studios, 1200 Park Avenue, Emeryville, CA 94608, USA
3 Department of Mathematics, University of Maryland, College Park, MD 20742-4015, USA 
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Garoufalidis, Stavros; Goerner, Matthias; Zickert, Christian. Gluing equations for PGL(n, ℂ)–representations of 3–manifolds. Algebraic and Geometric Topology, Tome 15 (2015) no. 1, pp. 565-622. doi: 10.2140/agt.2015.15.565

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