The baseleaf preserving mapping class group of the universal hyperbolic solenoid
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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 $\pi _1(X,*)$.References
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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
- Received by editor(s): December 4, 2000
- Received by editor(s) in revised form: July 31, 2003
- Published electronically: April 27, 2004
- © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 357 (2005), 1829-1858
- MSC (2000): Primary 57M60, 20F38; Secondary 30F60
- DOI: https://doi.org/10.1090/S0002-9947-04-03472-5
- MathSciNet review: 2115078
Dedicated: Dedicated to the memory of Subhashis Nag