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 Rep
max(Γ
g
,
G) in the representation variety Rep(Γ
g
,
G). Rep
max(Γ
g
,
G) is a union of connected components with similar properties as Teichmüller space
T(Sg) = Repmax(Gg, PSL(2,\mathbbR))\mathcal{T}(\Sigma_g) = {\rm Rep}_{\max}(\Gamma_g, {\rm PSL}(2,\mathbb{R})). We prove that the mapping class group
ModSgMod_{\Sigma_g} acts properly on Rep
max(Γ
g
,
G) when
G = Sp(2n,\mathbbR)G= {\rm Sp}(2n,\mathbb{R}), SU(
n,
n), SO*(4
n), 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