On graded quotient modules of mapping class groups of surfaces (Q1895073)

The Torelli subgroup of \(\Gamma_{g,n}\), the mapping class group of a compact orientable surface of genus \(g\) with \(n\) punctures, admits a natural weight filtration \(\{\Gamma_{g,n}(m)\}\), \(m \geq 1\), and the group \(\text{Sp} (2g,\mathbb{Q}) \times S_n\) (where Sp and \(S_n\) denote symplectic and symmetric groups, respectively) acts naturally on the graded quotients \(\text{gr}(g,n,m) = \Gamma_{g,n} (m)/\Gamma_{g,n} (m+1) \otimes \mathbb{Q}\), which are finite dimensional vector spaces over \(\mathbb{Q}\). The authors determine the \(\text{Sp} (2g, \mathbb{Q}) \times S_n\) module structure of \(\text{gr} (g,n,m)\) for \(m = 1,2,3\). For general \(m\), the authors construct explicit elements in \(\text{gr}(g,n,1)\) and \(\text{gr}(g,n,2)\) that yield \(\text{Sp} (2g, \mathbb{Q})\) -- irreducible components of \(\text{gr} (g,n,m)\).