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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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  • 1. R. Baer, Isotopien von Kurven auf orientierbaren, geschlossenen Flächen und ihr Zusammenhang mit der topologischen Deformation der Flächen, J. Reine Angew. Math. 159 (1928), 101-116.
  • 2. I. Biswas, M. Mitra, and S. Nag, Thurston boundary of the Teichmüller spaces and the commensurability modular group, Conform. Geom. Dyn. 3 (1999), 50-66. MR 2000b:32030
  • 3. I. Biswas and S. Nag, Commensurability automorphism groups and infinite constructions in Teichmüller theory, C. R. Acad. Sci. Paris Ser. I Math. 327 (1998), no. 1, 35-40. MR 99i:32026
  • 4. I. Biswas and S. Nag, Jacobians of Riemann surfaces and the Sobolev space $H\sp {1/2}$ on the circle, Math. Res. Lett. 5 (1998), no. 3, 281-292. MR 99g:58009
  • 5. I. Biswas and S. Nag, Weil-Petersson geometry and determinant bundles on inductive limits of moduli spaces, Lipa's legacy (New York, 1995), 51-80, Contemp. Math., 211, Amer. Math. Soc., Providence, RI, 1997. MR 99m:14050
  • 6. I. Biswas and S. Nag, Limit constructions over Riemann surfaces and their paramater spaces, and the commensurability group actions, Selecta Math. (N.S.) 6 (2000), no. 2, 185-224. MR 2002f:32026
  • 7. I. Biswas, S. Nag, and D. Sullivan, Determinant bundles, Quillen metrics and Mumford isomorphisms over the universal commensurability Teichmüller space, Acta Math. 176 (1996), 145-169. MR 97h:32030
  • 8. A. J. Casson and S. A. Bleiler, Automorphisms of surfaces after Nielsen and Thurston, London Math. Soc. Student Texts, vol. 9, Cambridge Univ. Press, Cambridge and New York, 1988. MR 89k:57025
  • 9. A. Douady and C. J. Earle, Conformally natural extension of homeomorphisms of the circle, Acta Math. 157 (1986), no. 1-2, 23-48. MR 87j:30041
  • 10. D. B. A. Epstein, Curves on $2$-manifolds and isotopies, Acta Math. 115 (1966), 83-107. MR 35:4938
  • 11. W. Floyd, Group completions and limit sets of Kleinian groups, Invent. Math. 57 (1980), no. 3, 205-218. MR 81e:57002
  • 12. R. H. Fox, On Fenchel's conjecture about $F$-groups, Mat. Tidsskr. B (1952), 61-65. MR 14:843c
  • 13. A. G. Kurosh, Theory of Groups, Vol. II, Chelsea, New York, 1956. MR 18:188f
  • 14. A. I. Mal'cev, Nilpotent torsion-free groups, Izvestiya Akad. Nauk SSSR. 13 (1949), 201-212. MR 10:507e
  • 15. F. Menegazzo and M. J. Tomkinson, Groups with trivial virtual automorphism group, Israel J. Math. 71 (1990), no. 3, 297-308. MR 92c:20072
  • 16. S. Morita, Characteristic classes of surface bundles, Bull. AMS 11 (1984), no. 2, 386-388. MR 85j:55032
  • 17. D. Mumford, Tata lectures on theta. I (with C. Musili, M. Nori, E. Previato and M. Stillman), Progress in Mathematics, vol. 28, Birkhäuser Boston, Inc., Boston, MA, 1983. MR 85h:14026
  • 18. S. Nag, Mathematics in and out of string theory, Topology and Teichm|ller spaces (Katinkulta, 1995), 187-220, World Sci. Publishing, River Edge, NJ, 1996. MR 99j:58221
  • 19. J. Nielsen, Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen, Acta. Math. 50 (1927), 189-358.
  • 20. C. Odden, The virtual automorphism group of the fundamental group of a closed surface, Ph.D. thesis, Duke University, 1997.
  • 21. D. Sullivan, Linking the universalities of Milnor-Thurston, Feigenbaum and Ahlfors-Bers, Topological methods in modern mathematics (Stony Brook, NY, 1991), Publish or Perish, Houston, TX, 1993, pp. 543-564. MR 94c:58060
  • 22. R. J. Zimmer, Ergodic theory and semisimple groups, Monographs in Mathematics, vol. 81, Birkhäuser Verlag, Basel and Boston, 1984. MR 86j:22014

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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

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

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