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

 

Discrete groups and the complex contact geometry of $ Sl(2,\mathbb{C})$


Author: Brendan Foreman
Journal: Trans. Amer. Math. Soc. 362 (2010), 4191-4200
MSC (2000): Primary 32M05; Secondary 11E57, 30F40, 57R17
Published electronically: February 12, 2010
MathSciNet review: 2608401
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Abstract: We investigate the vertical foliation of the standard complex contact structure on $ \Gamma\setminus Sl(2,\mathbb{C})$, where $ \Gamma$ is a discrete subgroup. We find that, if $ \Gamma$ is nonelementary, the vertical leaves on $ {\Gamma}\setminus Sl(2,\mathbb{C})$ are holomorphic but not regular. However, if $ \Gamma$ is Kleinian, then $ \Gamma\setminus Sl(2,\mathbb{C})$ contains an open, dense set on which the vertical leaves are regular, complete and biholomorphic to $ \mathbb{C}^*$. If $ \Gamma$ is a uniform lattice, the foliation is nowhere regular, although there are both infinitely many compact and infinitely many nonclosed leaves.


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Additional Information

Brendan Foreman
Affiliation: Department of Mathematics, John Carroll University, University Heights, Ohio 44118

DOI: http://dx.doi.org/10.1090/S0002-9947-10-04972-X
PII: S 0002-9947(10)04972-X
Received by editor(s): March 20, 2008
Published electronically: February 12, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.