Discrete representations in the character variety of a surface group

2017 Fiscal Year Final Research Report

• Project/Area Number: 26800038
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Kitami Institute of Technology (2016-2017); Kyoto University (2014-2015)
• Principal Investigator: Kabaya Yuichi 北見工業大学, 工学部, 准教授 (70551703)
• Project Period (FY): 2014-04-01 – 2018-03-31
• Keywords: クライン群 / 指標多様体 / 擬フックス群 / 双曲幾何学 / 複素射影構造 / 曲面群 / 写像類群

Outline of Final Research Achievements

For a hyperbolic surface S, the set of PSL(2,C)-representations of the fundamental group of S up to conjugation is called the character variety X(S). For simplicity, we assume that S is a once-punctured torus. We studied the subset of the character variety consisting of discrete faithful representations, or its open dense subset consisting of quasi-Fuchsian representations. In particular, we studied the slice of X(S) obtained by fixing the (complex) length of a simple closed curve. It is known that the set of quasi-Fuchsian representations in the slice consists of one connected component if the length is short, but there are more than two components if the length is long. In this research porject, we showed that these components are characterized in terms of Goldman's claasification of the complex projective structures with quasi-Fuchsian holonomy. As a corollary, we showed that there are infinitely many connected components in the slice if it has more than two components.

Free Research Field

双曲幾何学