On graded quotient modules of mapping class groups of surfaces
type: Item
instance of: scholarly article
title: On graded quotient modules of mapping class groups of surfaces
zbMATH Open document ID: 0851.57014
DOI: 10.1007/BF02783208
author: Mamoru Asada
author: Hiroaki Nakamura
published in: Israel Journal of Mathematics
publication date: 1996-11-24T00:00:00Z
review text: 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 \(\text{gr} (g,n,m)\).
Mathematics Subject Classification ID: 57M99
Mathematics Subject Classification ID: 20F38
zbMATH DE Number: 784913
zbMATH Keywords: Torelli subgroup
zbMATH Keywords: mapping class group
zbMATH Keywords: surface
reviewed by: Wolfgang Heil
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02783208
OpenAlex ID: W1991798331
description: scientific article; zbMATH DE number 784913