Abstract
Let Γ g be the fundamental group of a closed oriented Riemann surface Σ g , g ≥ 2, and let G be a simple Lie group of Hermitian type. The Toledo invariant defines the subset of maximal representations Repmax(Γ g , G) in the representation variety Rep(Γ g , G). Repmax(Γ g , G) is a union of connected components with similar properties as Teichmüller space \(\mathcal{T}(\Sigma_g) = {\rm Rep}_{\max}(\Gamma_g, {\rm PSL}(2,\mathbb{R}))\). We prove that the mapping class group \(Mod_{\Sigma_g}\) acts properly on Repmax(Γ g , G) when \(G= {\rm Sp}(2n,\mathbb{R})\), SU(n,n), SO*(4n), Spin(2,n).
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Wienhard, A. The Action of the Mapping Class Group on Maximal Representations. Geom Dedicata 120, 179–191 (2006). https://doi.org/10.1007/s10711-006-9079-7
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DOI: https://doi.org/10.1007/s10711-006-9079-7
Keywords
- Mapping class group
- Modular group
- Representation variety
- Maximal representations
- Toledo invariant
- Teichmüller space