We consider a surface Σ of genus g≥3 , either closed or with exactly one puncture. The mapping class group Γ of Σ acts symplectically on the abelian moduli space M=Hom(π 1 (Σ),U(1))=Hom(H 1 (Σ),U(1)) , and hence both L 2 (M) and C ∞ (M) are modules over Γ . In this paper, we prove that both the cohomology groups H 1 (Γ,L 2 (M)) and H 1 (Γ,C ∞ (M)) vanish.
Quantum Topology, 2012, Vol 3, Issue 3/4, p. 359-376
mapping class groups; group cohomology; moduli spaces; property (T)