An action of subgroups of mapping class groups on polynomial algebras
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.
scientific article; zbMATH DE number 5132968
scientific article