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 $\operatorname{Out}(G)$ 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.