Residual $p$ properties of mapping class groups and surface groups

Abstract:

Let $\mathcal {M}(\Sigma , \mathcal {P})$ be the mapping class group of a punctured oriented surface $(\Sigma ,\mathcal {P})$ (where $\mathcal {P}$ may be empty), and let $\mathcal {T}_p(\Sigma ,\mathcal {P})$ be the kernel of the action of $\mathcal {M} (\Sigma , \mathcal {P})$ on $H_1(\Sigma \setminus \mathcal {P}, \mathbb {F}_p)$. We prove that $\mathcal {T}_p( \Sigma ,\mathcal {P})$ is residually $p$. In particular, this shows that $\mathcal {M} (\Sigma ,\mathcal {P})$ is virtually residually $p$. For a group $G$ we denote by $\mathcal {I}_p(G)$ the kernel of the natural action of $\operatorname {Out}(G)$ on $H_1(G,\mathbb {F}_p)$. In order to achieve our theorem, we prove that, under certain conditions ($G$ is conjugacy $p$-separable and has Property A), the group $\mathcal {I}_p(G)$ is residually $p$. The fact that free groups and surface groups have Property A is due to Grossman. The fact that free groups are conjugacy $p$-separable is due to Lyndon and Schupp. The fact that surface groups are conjugacy $p$-separable is, from a technical point of view, the main result of the paper.