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

 

The baseleaf preserving mapping class group of the universal hyperbolic solenoid


Author: Chris Odden
Journal: Trans. Amer. Math. Soc. 357 (2005), 1829-1858
MSC (2000): Primary 57M60, 20F38; Secondary 30F60
Published electronically: April 27, 2004
MathSciNet review: 2115078
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Abstract: Given a closed surface $X$, the covering solenoid $\mathbf{X}_\infty$ is by definition the inverse limit of all finite covering surfaces over $X$. If the genus of $X$ is greater than one, then there is only one homeomorphism type of covering solenoid; it is called the universal hyperbolic solenoid. In this paper we describe the topology of $\Gamma(\mathbf{X}_\infty)$, the mapping class group of the universal hyperbolic solenoid. Central to this description is the notion of a virtual automorphism group. The main result is that there is a natural isomorphism of the baseleaf preserving mapping class group of $\mathbf{X}_\infty$ onto the virtual automorphism group of $\pi_1(X,*)$. This may be regarded as a genus independent generalization of the theorem of Dehn, Nielsen, Baer, and Epstein that the pointed mapping class group $\Gamma(X,*)$ is isomorphic to the automorphism group of $\pi_1(X,*)$.


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Additional Information

Chris Odden
Affiliation: Department of Mathematics and Computer Science, Amherst College, Amherst, Massachusetts 01002
Address at time of publication: Department of Mathematics, Phillips Academy, Andover, Massachusetts 01810
Email: ctodden@andover.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03472-5
PII: S 0002-9947(04)03472-5
Keywords: Virtual automorphism, mapping class group, Teichm\"uller theory
Received by editor(s): December 4, 2000
Received by editor(s) in revised form: July 31, 2003
Published electronically: April 27, 2004
Dedicated: Dedicated to the memory of Subhashis Nag
Article copyright: © Copyright 2004 American Mathematical Society