On the character variety of the four-holed sphere
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 \(\mathrm{PSL}_2(\mathbb Z)\)-action on the latter. They also use the same ideas to study representations in the real locus of the character variety, corresponding to morphisms from \(F_3\) to \(\mathrm{SL}_2(\mathbb{R})\) or \(\mathrm{SU}(2)\).
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