6533b86dfe1ef96bd12c9dfc

RESEARCH PRODUCT

Residual 𝑝 properties of mapping class groups and surface groups

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Let M ( Σ , P ) \mathcal {M}(\Sigma , \mathcal {P}) be the mapping class group of a punctured oriented surface ( Σ , P ) (\Sigma ,\mathcal {P}) (where P \mathcal {P} may be empty), and let T p ( Σ , P ) \mathcal {T}_p(\Sigma ,\mathcal {P}) be the kernel of the action of M ( Σ , P ) \mathcal {M} (\Sigma , \mathcal {P}) on H 1 ( Σ ∖ P , F p ) H_1(\Sigma \setminus \mathcal {P}, \mathbb {F}_p) . We prove that T p ( Σ , P ) \mathcal {T}_p( \Sigma ,\mathcal {P}) is residually p p . In particular, this shows that M ( Σ , P ) \mathcal {M} (\Sigma ,\mathcal {P}) is virtually residually p p . For a group G G we denote by I p ( G ) \mathcal {I}_p(G) the kernel of the natural action of Out ⁡ ( G ) \operatorname {Out}(G) on H 1 ( G , F p ) H_1(G,\mathbb {F}_p) . In order to achieve our theorem, we prove that, under certain conditions ( G G is conjugacy p p -separable and has Property A), the group I p ( G ) \mathcal {I}_p(G) is residually p p . The fact that free groups and surface groups have Property A is due to Grossman. The fact that free groups are conjugacy p p -separable is due to Lyndon and Schupp. The fact that surface groups are conjugacy p p -separable is, from a technical point of view, the main result of the paper.