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# Residual p Properties of Mapping Class Groups and Surface Groups

Luis Paris
Transactions of the American Mathematical Society
Vol. 361, No. 5 (May, 2009), pp. 2487-2507
Stable URL: http://www.jstor.org/stable/40302863
Page Count: 21
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## Abstract

Let . M(∑,P) be the mapping class group of a punctured oriented surface (∑P) (where P may be empty), and let $T_p (\Sigma ,{\rm{P}})$ be the kernel of the action of M(∑,P) on $H_1 (\Sigma \backslash {\rm{P}},F_p )$ We prove that $T_p (\Sigma ,{\rm P})$ is residually p. In particular, this shows that M(∑,P) is virtually residually p. For a group G we denote by $I_p (G)$ the kernel of the natural action of Out(G) on $H_1 (G,F_p ).$. In order to achieve our theorem, we prove that, under certain conditions (G is conjugacy p-separable and has Property A), the group $I_p (G)$ 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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