Abstract
The punctured solenoid \({\mathcal H}\) plays the role of an initial object for the category of punctured surfaces with morphisms given by finite covers branched only over the punctures. The (decorated) Teichmüller space of \({\mathcal H}\) is introduced, studied, and found to be parametrized by certain coordinates on a fixed triangulation of \({\mathcal H}\). Furthermore, a point in the decorated Teichmüller space induces a polygonal decomposition of \({\mathcal H}\) itself giving a combinatorial description of the decorated Teichmüller space. This is used to obtain a non-trivial set of generators of the modular group of \({\mathcal H}\), and each word in these generators admits a normal form. There is furthermore a non-degenerate modular group invariant two form on the Teichmüller space of \({\mathcal H}\). All of this structure is in perfect analogy with that of the decorated Teichmüller space of a punctured surface of finite type.
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References
Ahlfors L.V. (1966). Lectures on Quasiconformal Mappings. D. Van Nostrand Company, Inc., Princeton
Ahlfors L.V. and Bers L. (1960). Riemann’s Mapping Theorem for Variable Metrics. Ann. Math. 72(2): 385–404
Biswas, I., Nag, S.: Weill-Peterson geometry and determinant bundles on inductive limits of moduli spaces, Lipa’s Legacy (New york, 1995), pp. 51–80, Contemp. Math. 211, Amer. Math. Soc., Providence, RI (1997)
Biswas I. and Nag S. (2000). Limit constructions over Riemann surfaces and their parameter spaces and the commensurability group action. Sel. math. New ser. 6: 185–224
Bonnot, S., Penner, R.C., Šarić, D.: A presentation for the baseleaf preserving mapping class group of the punctured solenoid. Alg. Geom. Top. (math.DS/0608066 to appear)
Candel A. (1993). Uniformization of surface laminations. Ann. Sci. École Norm. Sup. 26(4): 489–516
Fock, V.V., Chekhov, L.: Quantum Teichmüller spaces (Russian) Teoret. Mat. Fiz. 120(3), 511–528 (1999). [Translation in Theoret. and Math. Phys. 120(3), 1245–1259]
Douady A. and Earle C.J. (1986). Conformally natural extension of homeomorphisms of the circle. Acta Math. 157(1–2): 23–48
Epstein D., Marden A. and Markovic V. (2004). Quasiconformal homeomorphisms and the convex hull boundary. Ann. Math. (2) 159(no. 1): 305–336
Earle, C., McMullen, C.: Quasiconformal isotopies. In: Holomorphic Functions and Moduli, vol. I, pp. 143–154, (Berkeley, CA, 1986). Math. Sci. Res. Inst. Publ., no. 10, Springer, New York (1988)
Fock, V., Goncharov, A.: Moduli spaces of local systems and higher Teichmüller theory. Publ. Math. Inst. Hautes tudes Sci. No. 103 (2006), pp. 1–211
Igusa, K.: Graph cohomology and Kontsevich cycles. math.AT/0303157
Kashaev R.M. (1998). Quantization of Teichmüller spaces and the quantum dilogarithm. Lett. Math. Phys. 43(2): 105–115; q-alg/9705021
Lochak, P., Schneps, L.: The universal Ptolemy-Teichm ller groupoid. In: Geometric Galois actions, 2, pp. 325–347. London Math. Soc. Lecture Note Ser., vol. 243, Cambridge Univ. Press, Cambridge (1997)
Margulis, G.A.: Discrete groups of motions of manifolds of non-positive curvature. In: Proc. Int. Cong. Math., vol. 2, pp. 21–34. Vancouver (1974)
Markovic V. and Šarić D. (2006). The Teichmüller Mapping Class Group of the Universal Hyperbolic Solenoid. Trans. Amer. Math. Soc. 358(6): 2637–2650
Nag S. and Sullivan D. (1995). Teichmüller theory and the universal period mappings via quantum calculus and the H1/2 space of the circle. Osaka. J. Math. 32: 1–34
Odden, C.: Virtual automorphism group of the fundamental group of a closed surface. Ph.D. Thesis, Duke University, Durham (1997)
Odden C. (2004). The baseleaf preserving mapping class group of the universal hyperbolic solenoid. Trans. Amer. Math Soc. 357: 1829–1858
Penner R.C. (1993). Universal constructions in Teichmüller theory. Adv. Math. 98: 143–215
Penner R.C. (1987). The decorated Teichmüller space of punctured surfaces. Comm. Math. Phys. 113: 299–339
Penner R.C. (1992). The Weil-Petersson volumes. J. Differentical Geom. 38: 559–608
Penner, R.C.: The universal Ptolemy group and its completions. In: Lochak, P., Schneps, L. (eds.) Geometric Galois Actions II, London Math Society Lecture Notes 243, Cambridge University Press (1997)
Šarić, D.: On Quasiconformal Deformations of the Universal Hyperbolic Solenoid. (To appear J. Anal. Math)
Sullivan, D.: Linking the universalities of Milnor-Thurston, Feigenbaum and Ahlfors-Bers. In: Goldberg, L., Phillips, A. (eds.) Milnor Festschrift, Topological methods in modern mathematics, pp. 543–563. Publish or Perish (1993)
Thurston, W.: Earthquakes in two-dimensional hyperbolic geometry. In: Epstein, D.B.A., (ed.) Low-dimensional Topology and Kleinian Groups. Warwick and Durham, 1984. L.M.S. Lecture Note Series, vol. 112, pp. 91–112. Cambridge University Press, Cambridge (1986)
Thurston, W.: The geometry and topology of 3-manifolds. Princeton University Lecture Notes, online at http://www.msri.org/publications/books/gt3m. Accessed March 2002.
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Dedicated to the memory of Subhashis Nag.
Dragomir Šarić was partially supported by NSF grant DMS 0505652.
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Penner, R.C., Šarić, D. Teichmüller theory of the punctured solenoid. Geom Dedicata 132, 179–212 (2008). https://doi.org/10.1007/s10711-007-9226-9
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DOI: https://doi.org/10.1007/s10711-007-9226-9