Teichmüller mapping class group of the universal hyperbolic solenoid

Markovic, Vladimir; Šarić, Dragomir

doi:10.1090/S0002-9947-05-03823-7

Teichmüller mapping class group of the universal hyperbolic solenoid
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by Vladimir Markovic and Dragomir Šarić PDF

Trans. Amer. Math. Soc. 358 (2006), 2637-2650 Request permission

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 $MCG_{BLP}(H_\infty )$.

References

Lipman Bers, Universal Teichmüller space, Analytic methods in mathematical physics (Sympos., Indiana Univ., Bloomington, Ind., 1968) Gordon and Breach, New York, 1970, pp. 65–83. MR 0349988
Indranil Biswas and Subhashis Nag, Weil-Petersson geometry and determinant bundles on inductive limits of moduli spaces, Lipa’s legacy (New York, 1995) Contemp. Math., vol. 211, Amer. Math. Soc., Providence, RI, 1997, pp. 51–80. MR 1476981, DOI 10.1090/conm/211/02814
Indranil Biswas and Subhashis Nag, Limit constructions over Riemann surfaces and their parameter spaces, and the commensurability group actions, Selecta Math. (N.S.) 6 (2000), no. 2, 185–224. MR 1816860, DOI 10.1007/PL00001388
Alberto Candel, Uniformization of surface laminations, Ann. Sci. École Norm. Sup. (4) 26 (1993), no. 4, 489–516. MR 1235439
Adrien Douady and Clifford J. Earle, Conformally natural extension of homeomorphisms of the circle, Acta Math. 157 (1986), no. 1-2, 23–48. MR 857678, DOI 10.1007/BF02392590
Clifford J. Earle and Curt McMullen, Quasiconformal isotopies, Holomorphic functions and moduli, Vol. I (Berkeley, CA, 1986) Math. Sci. Res. Inst. Publ., vol. 10, Springer, New York, 1988, pp. 143–154. MR 955816, DOI 10.1007/978-1-4613-9602-4_{1}2
Étienne Ghys, Laminations par surfaces de Riemann, Dynamique et géométrie complexes (Lyon, 1997) Panor. Synthèses, vol. 8, Soc. Math. France, Paris, 1999, pp. ix, xi, 49–95 (French, with English and French summaries). MR 1760843
Steven P. Kerckhoff, The Nielsen realization problem, Ann. of Math. (2) 117 (1983), no. 2, 235–265. MR 690845, DOI 10.2307/2007076
G. A. Margulis, Discrete groups of motions of manifolds of nonpositive curvature, Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974) Canad. Math. Congress, Montreal, Que., 1975, pp. 21–34 (Russian). MR 0492072
Subhashis Nag and Dennis Sullivan, Teichmüller theory and the universal period mapping via quantum calculus and the $H^{1/2}$ space on the circle, Osaka J. Math. 32 (1995), no. 1, 1–34. MR 1323099
C. Odden, Virtual automorphism group of the fundamental group of a closed surface, Ph.D. Thesis, Duke University, Durham, 1997.
D. Šarić, On Quasiconformal Deformations of the Universal Hyperbolic Solenoid, preprint, available at: http://math.usc.edu/$\sim$ saric.
Dennis 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 1215976