Parametrization of PSL(2,C)-representations of surface groups

2011/10/31 by Yuichi Kabaya, Kabaya, Yuichi · 1 citation
Mathematics · #30F60 #32G15 #57M50 #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.GT #msc:30F60 #msc:32G15 #msc:57M50

paper · pdf · doi:10.48550/arxiv.1110.6674

55 pages, 31 figures; minor corrections (no change in formulas), accepted for publication in Geometriae Dedicata

openalex publication_date 2011/10/31 · arxiv created 2013/05/03 · arxiv updated 2013/05/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For an oriented surface of genus g with b boundary components, we construct a rational map from a subset of C^6g-6+3b onto an open algebraic subset of the PSL(2,C)-character variety as an analogue of the Fenchel-Nielsen coordinates. After taking the quotient by an action of a finite group, we obtain a parametrization of a subset of the PSL(2,C)-character variety, and similarly for the SL(2,C)-character variety. We can systematically calculate a set of matrix generators by rational functions of the parameters. We give transformation formulae under elementary moves of pants decompositions.

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• Peripheral subgroups of Kleinian groups

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