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

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 ${\rm Out} (G)$ 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.