On the character variety of the four-holed sphere
type: Item
instance of: scholarly article
title: On the character variety of the four-holed sphere
zbMATH Open document ID: 1330.57025
DOI: 10.4171/GGD/326
author: Sara Maloni
author: Frederic Palesi
author: Ser Peow Tan
published in: Groups, Geometry, and Dynamics
publication date: 2015-10-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/1304.5770
review text: Let \(S\) be the four-holed sphere. The fundamental group \(\pi_1(S)\) of \(S\) is the group generated by the four boundary loops \(\alpha,\beta,\gamma,\delta\) which satisfy the relation \(\alpha\beta\gamma\delta = 1\). It is thus isomorphic to the free group \(F_3\) on three generators. The \(\mathrm{SL}_2(\mathbb C)\)-character variety \(\mathcal X\) is the quotient of the algebraic variety \(\mathbb C^3 = \mathrm{Hom}(F_3, \mathrm{SL}_2(\mathbb C))\) by the \(\mathrm{SL}_2(\mathbb C)\)-action by conjugation. The identification of \(F_3\) with \(\pi_1(S)\) allows to define an action of the mapping class group \(\mathcal{MCG}(S)\) on \(\mathcal X\) by precomposition. This action preserves the subsets \(\mathcal X_{a,b,c,d}\) defined by \(\mathrm{tr}(\alpha) = a\), etc. The main result in the present paper is that each set \(\mathcal X_{\mathbf\tau}\) (where \(\mathbf \tau = (a,b,c,d)\)) contains an open subset on which the mapping class group acts properly discontinuously. This subset is defined by the following conditions: {\parindent=6mm \begin{itemize} \item[-] No simple closed curve on \(S\) has an elliptic image; \item [-] There are at most finitely many simple closed curves on \(S\) whose images' traces are less (in absolute value) than a constant (depending on \(\mathbf \tau\)). \end{itemize}} The proof proceeds by identifying the character variety with the set of ``Markov maps'' (introduced in this context by \textit{B. H. Bowditch} [Proc. Lond. Math. Soc. (3) 77, No. 3, 697--736 (1998; Zbl 0928.11030)]) from \(\mathbb Q\cup \infty\) to \(\mathbb C\) (the latter being identified with the set of simple closed curves by a result of \textit{L. Keen} and \textit{C. Series} [Proc. Lond. Math. Soc. (3) 69, No. 1, 72--90 (1994; Zbl 0807.30031)]). These maps have a combinatorial description in terms of the dual tree of the Farey tesselation of the hyperbolic plane, and the mapping class group action on the character variety is related to the \(\mathrm{PSL}_2(\math…
reviewed by: Jean Raimbault
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 37A15
zbMATH DE Number: 6496566
zbMATH Keywords: character variety
zbMATH Keywords: mapping class group
zbMATH Keywords: Markov maps
zbMATH Keywords: domain of discontinuity
MaRDI profile type: Publication
arXiv ID: 1304.5770
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description: scientific article