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

 

Teichmüller mapping class group of the universal hyperbolic solenoid


Authors: Vladimir Markovic and Dragomir Saric
Journal: Trans. Amer. Math. Soc. 358 (2006), 2637-2650
MSC (2000): Primary 30F60; Secondary 32G05, 32G15, 37F30
Published electronically: October 31, 2005
MathSciNet review: 2204048
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Abstract: We show that the homotopy class of a quasiconformal self-map of the universal hyperbolic solenoid $ H_\infty$ is the same as its isotopy class and that the uniform convergence of quasiconformal self-maps of $ H_\infty$ to the identity forces them to be homotopic to conformal maps. We identify a dense subset of $ \mathcal{T}(H_\infty )$ such that the orbit under the base leaf preserving mapping class group $ MCG_{BLP}(H_\infty)$ of any point in this subset has accumulation points in the Teichmüller space $ \mathcal{T}(H_\infty )$. Moreover, we show that finite subgroups of $ MCG_{BLP}(H_\infty )$ are necessarily cyclic and that each point of $ \mathcal{T}(H_\infty)$ has an infinite isotropy subgroup in $ MCG_{BLP}(H_\infty )$.


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Additional Information

Vladimir Markovic
Affiliation: Institute of Mathematics, University of Warwick, CV4 7AL Coventry, United Kingdom
Email: markovic@maths.warwick.ac.uk

Dragomir Saric
Affiliation: Institute of Mathematics, State University of New York, Stony Brook, New York 11794-3660
Email: saric@math.sunysb.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-05-03823-7
PII: S 0002-9947(05)03823-7
Received by editor(s): July 22, 2004
Published electronically: October 31, 2005
Article copyright: © Copyright 2005 American Mathematical Society