, Volume 90, Issue 1-3, pp 93-113

On graded quotient modules of mapping class groups of surfaces

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Abstract

Let Γ g, n be the mapping class group of a compact Riemann surface of genusg withn points preserved (2−2gn<0,g≥1,n≥0). The Torelli subgroup of Γ g, n has a natural weight filtration {Γg, n(m)} m≥1. Each graded quotient gr m Γ g, n ⊗ ℚ (m≥1) is a finite dimensional vector space over ℚ on which the group Sp(2g, ℚ)×S n naturally acts.

In this paper, we have determined the Sp(2g, ℚ)×S n module structure of gr m Γ g, n ⊗ ℚ for 1≤m≤3. This includes a verification of an expectation by S. Morita. Also, for generalm, we have identified a certain Sp(2g, ℚ)-irreducible component of gr m Γ g, n ⊗ ℚ by constructing explicitly elements in these modules.