Original Paper

Geometriae Dedicata

, Volume 120, Issue 1, pp 179-191

First online:

The Action of the Mapping Class Group on Maximal Representations

  • Anna WienhardAffiliated withSchool of Mathematics, Institute for Advanced StudyDepartment of Mathematics, University of Chicago Email author 

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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 g , G) in the representation variety Rep(Γ g , G). Repmax 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 g , G) when \(G= {\rm Sp}(2n,\mathbb{R})\), SU(n,n), SO*(4n), Spin(2,n).

Keywords

Mapping class group Modular group Representation variety Maximal representations Toledo invariant Teichmüller space

Mathematics Subject Classifications (2000)

Primary 20H10 Secondary 32M15 32G15