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The modular action on \(\mathrm{PSL}_2(\mathbb{R})\)-characters in genus 2. (English) Zbl 1353.37056

Summary: We explore the dynamics of the action of the mapping class group in genus \(2\) on the \(\mathrm{PSL}_{2}(\mathbb{R})\)-character variety. We prove that this action is ergodic on the connected components of Euler class \(\pm 1\), as it was conjectured by Goldman. In the connected component of Euler class \(0\) there are two invariant open subsets; on one of them the action is ergodic. In this process we give a partial answer to a question posed by Bowditch.

MSC:

37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
57M05 Fundamental group, presentations, free differential calculus
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
30F60 Teichmüller theory for Riemann surfaces
58D29 Moduli problems for topological structures
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References:

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