Residual $p$ properties of mapping class groups and surface groups
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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.References
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Additional Information
- Luis Paris
- Affiliation: Institut de Mathématiques de Bourgogne, UMR 5584 du CNRS, Université de Bourgogne, B.P. 47870, 21078 Dijon cedex, France
- Email: lparis@u-bourgogne.fr
- Received by editor(s): April 2, 2007
- Published electronically: November 3, 2008
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 361 (2009), 2487-2507
- MSC (2000): Primary 20F38; Secondary 20E26, 20F14, 20F34, 57M99
- DOI: https://doi.org/10.1090/S0002-9947-08-04573-X
- MathSciNet review: 2471926