Geometriae Dedicata

, Volume 120, Issue 1, pp 179–191

The Action of the Mapping Class Group on Maximal Representations

Original Paper

Abstract

Let Γg be the fundamental group of a closed oriented Riemann surface Σg, g ≥ 2, and let G be a simple Lie group of Hermitian type. The Toledo invariant defines the subset of maximal representations Repmaxg, G) in the representation variety Rep(Γg, G). Repmaxg, 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 Repmaxg, 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 

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Copyright information

© Springer Science+Business Media B.V. 2006

Authors and Affiliations

  1. 1.School of MathematicsInstitute for Advanced StudyPrincetonUSA
  2. 2.Department of MathematicsUniversity of ChicagoChicagoUSA

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