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Congruence kernels around affine curves

  • Richard Peabody Kent IV EMAIL logo
Veröffentlicht/Copyright: 11. April 2014

Abstract

Let S be a smooth affine algebraic curve, and let S˚ be the Riemann surface obtained by removing a point from S. We provide evidence for the congruence subgroup property of mapping class groups by showing that the congruence kernel

ker Mod (S˚)^ Out (π1(S˚)^)

lies in the centralizer of every braid in Mod (S˚). As a corollary, we obtain a new proof of Asada's theorem that the congruence subgroup property holds in genus one. We also obtain simple-connectivity of Boggi's procongruence curve complex 𝒞ˇ(S˚) when S is affine, and a new proof of Matsumoto's theorem that the congruence kernel depends only on the genus in the affine case.

Funding source: NSF

Award Identifier / Grant number: postdoctoral fellowship

Funding source: NSF

Award Identifier / Grant number: DMS-1104871

The author thanks Marco Boggi, Tom Church, Jordan Ellenberg, Chris Leininger, Pierre Lochak, Ben McReynolds, Andy Putman, Gereon Quick, and Ben Wieland for many useful conversations. The author thanks the referees for their careful readings and many helpful comments. The author also thanks the Institute for Advanced Study and the Park City Math Institute, where some of this work was carried out.

Received: 2012-1-24
Revised: 2014-2-14
Published Online: 2014-4-11
Published in Print: 2016-4-1

© 2016 by De Gruyter

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