On the filtration of topological and pro-\(\ell\) mapping class groups of punctured Riemann surfaces
Let \(R_{g,n}\) be a Riemann surface of genus \(g\geq 0\) with \(n\geq 0\) punctures, and \(\Gamma_{g,n}\) be the pure mapping class group of \(R_{g,n}\). As is well known, \(\Gamma_{g,n}\) is isomorphic to a certain subgroup of the outer automorphism group of the topological fundamental group \(\pi_{g,n}\) of \(R_{g,n}\). The group \(\pi_{g,n}\) has a natural filtration \(\{\pi_{g,n}(m)\}^\infty_{m=1}\) called the weight filtration, which is introduced by T. Oda. This filtration naturally induces a filtration \(\{\Gamma_{g,n}[m]\}^\infty_{m=0}\) of \(\Gamma_{g,n}\). The aim of this paper is to serve some basic results about this filtration. In this paper, we assume that \(2-2g-n<0\), which is equivalent to that the group \(\pi_{g,n}\) is non-abelian. Recall that there exists a canonical exact sequence \(1\to\pi_{g,n-1} @>d_*>> \Gamma_{g,n} @>p_*>> \Gamma_{g,n-1} \to 1\). The homomorphism \(p_*\) is induced from ``forgetting'' the \(n\)-th puncture and the homomorphism \(d_*\) can be explicitly described by a result of Birman. Then, it can be shown that the homomorphisms \(d_*\) and \(p_*\) preserve the filtrations, hence we have a complex \[ 0\to\text{gr}(\pi_{g,n-1})\to\text{gr}\bigl(\Gamma_{g,n}[1]\bigr)\to\text{gr} \bigl(\Gamma_{g,n-1} [1]\bigr] \to 0. \tag{*} \] Here, each associated graded module gr(\ ) has a structure of a graded Lie algebra. Moreover, if \(g\geq 1\), the Siegel modular group \(\text{Sp}(2g;\mathbb{Z})\) naturally acts on them. Theorem A. If \(n\geq 2\), the complex (*) is an exact sequence of graded Lie algebras with \(\text{Sp}(2g;\mathbb{Z})\)-action. The second result is about the comparison of \(\Gamma_{g,n}\) with the pro-\(l\) mapping class group, \(l\) being a fixed prime number. Theorem B. (i) For each \(m\geq 1\), the \(\mathbb{Z}\)-module \(\text{gr}^m (\Gamma_{g,n})\) and the \(\mathbb{Z}_l\)-module \(\text{gr}^m (\overline \Gamma_{g,n})\) are free of finite rank and \(\text{gr}^m (\varphi_l)\) induces an injective homomorphism \[ \text{gr}^m (\Gamma_{g,n}) \otimes_\mathbb{Z} \mathbb{Z}_l\to \text{gr}^m (\overline \Gamma_{ g,n}). \tag{**} \] If \(g\geq 1\), this homomorphism is \(\text{Sp}(2g)\)-equivariant. (ii) Assume that \(g\neq 1\). Then we have \(\overline {\Gamma_{g,n} [m]}= \overline \Gamma_{g,n} [m]\) \((m\geq 0)\), hence the homomorphism (**) is an isomorphism. There are two crucial points for the proof of the results. One of them is that the graded Lie algebras \(\text{gr} (\pi_{g,n})\) and \(\text{gr} (\pi_{p,n})\otimes_\mathbb{Z} \mathbb{F}_p\) \((p\): prime) have trivial center. Another one, which is for the proof of Theorem B (ii), is some properties of the symplectic group, especially the congruence subgroup property of \(\text{Sp}(2g; \mathbb{Z})\) \((g\geq 2)\). In fact, in the case that \(g=1\) (and \(n=1)\), an analogous result does not hold.
scientific article; zbMATH DE number 894111