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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Embedding problems and open subgroups


Authors: David Harbater and Katherine Stevenson
Journal: Proc. Amer. Math. Soc. 139 (2011), 1141-1154
MSC (2010): Primary 14G17, 14H30, 20E18; Secondary 12E30, 14G32, 20F34
Published electronically: November 17, 2010
MathSciNet review: 2748409
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Abstract: We study the properties of the fundamental group of an affine curve over an algebraically closed field of characteristic $ p$ from the point of view of embedding problems. In characteristic zero the fundamental group is free, but in characteristic $ p$ it is not even $ \omega$-free. In this paper we show that it is ``almost $ \omega$-free'' in the sense that each finite embedding problem has a proper solution when restricted to some open subgroup. We also prove that embedding problems can always be properly solved over the given curve if suitably many additional branch points are allowed in locations that can be specified arbitrarily; this strengthens a result of the first author.


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Additional Information

David Harbater
Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6395
Email: harbater@math.upenn.edu

Katherine Stevenson
Affiliation: Department of Mathematics, California State University, Northridge, California 91330
Email: katherine.stevenson@csun.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10776-1
PII: S 0002-9939(2010)10776-1
Keywords: Fundamental group, affine curve, characteristic $p$, embedding problem, omega-free
Received by editor(s): December 6, 2009
Published electronically: November 17, 2010
Additional Notes: The first author was supported in part by NSF grant DMS-0901164.
The second author was supported in part by NSF grant IIS-0534984
Communicated by: Ted Chinburg
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.