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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Author: Luis Paris
Journal: Trans. Amer. Math. Soc. 361 (2009), 2487-2507
MSC (2000): Primary 20F38; Secondary 20E26, 20F14, 20F34, 57M99
Published electronically: November 3, 2008
MathSciNet review: 2471926
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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.


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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

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04573-X
PII: S 0002-9947(08)04573-X
Received by editor(s): April 2, 2007
Published electronically: November 3, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.