An action of subgroups of mapping class groups on polynomial algebras
type: Item
instance of: scholarly article
title: An action of subgroups of mapping class groups on polynomial algebras
zbMATH Open document ID: 1117.57018
author: Stephen P. Humphries
published in: Topology and its Applications
publication date: 2007-03-12T00:00:00Z
review text: Let \(F_n\) be a free group of rank \(n\) and let \(B_n\subset\text{Out}(F_n)\) be the braid group of \(n\) strands. \textit{W. Magnus} [Math. Z 170, 91--103 (1980; Zbl 0433.20033)] found that a certain action of the braid group on a subalgebra of a trace algebra, which is not a polynomial algebra, lifts to an action on a finitely generated polynomial algebra. If one thinks of braid groups as being mapping class groups of a planar surfaces with boundary, then it is natural to wonder whether there is an analogous action of the mapping class group of a non-planar surface with (or without) boundary on a polynomial algebra. The author gives some answers by obtaing an action of some subgroup of the mapping class group on a polynomial algebra. For example the following theorem is proved. Let \(\Sigma_{g,n,m}\) denote a surface of genus \(g<\infty\) with \(n<\infty\) punctures and \(m\geq 1\), \(m<\infty\), boundary components and let \({\mathcal M}_{g,n,m}\) denote the mapping class group of \(\Sigma_{g,n,m}\). Then, there is a finitely generated polynomial ring \(R_{g,n,m}\) (over a commutative ring with 1) and an ideal \(I_{g,n,m}\) of \(R_{g,n,m}\), such that there is a representation \({\mathcal M}_{g,n,m}\to\text{Aut}(R_{g,n,m}/I_{g,n, m})\). The author also investigates the invariant theory for the above action and various representations that this action affords.
reviewed by: Andrzej Szczepański
Mathematics Subject Classification ID: 57M99
Mathematics Subject Classification ID: 57N05
Mathematics Subject Classification ID: 20F36
Mathematics Subject Classification ID: 20F34
Mathematics Subject Classification ID: 20F29
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20F28
zbMATH DE Number: 5132968
zbMATH Keywords: mapping class group
zbMATH Keywords: polynomial algebra
zbMATH Keywords: Torelli subgroup
zbMATH Keywords: trace algebras
zbMATH Keywords: braid group
describes a project that uses: Magma
MaRDI profile type: Publication
full work available at URL: j.topol.2006.10.009
OpenAlex ID: W1977844570
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DOI: 10.1016/J.TOPOL.2006.10.009
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description: scientific article; zbMATH DE number 5132968