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The decorated Teichmüller space of punctured surfaces. (English) Zbl 0642.32012

Let \(F^ s_ g\) denote the genus \(g\) surface with \(s\) points removed, where \(2g-2+s>0,\) \(g\geq 0\), \(s\geq 1\). Let \({\mathcal T}^ s_ g\) be the corresponding Teichmüller space. The decorated Teichmüller space, introduced in the paper, is the total space of a fibration \(\phi:\tilde{\mathcal T}^ s_ g\to {\mathcal T}^ s_ g\), where the fiber over a point \(x\) is the space of \(s\)-tuples of horocycles about the \(s\) punctures of \(F^ s_ g\) (with respect to a hyperbolic metric on \(F^ s_ g\) representing \(x\)). Another central concept of the paper is that of an ideal triangulation and an ideal cell decomposition of \(F^ s_ g\). This is a system \(\Delta\) of disjoint pairwise nonisotopic arcs connecting the punctures of \(F^ s_ g\) such that every component of \(F^ s_ g\setminus \Delta\) is a triangle (respectively, a cell). The first of the main results of the paper gives: if the function \(f^*_ T(x)=\sup (1/N)\sum^{N}_{k=1}| f(T^{n_ k}x)|\) fails to be bounded in \(L^ p\) then the maximal function \(f^*_ S\) also fails to be bounded in \(L^ p\). A constructive proof is given using only the Rokhlin construction.
Reviewer: R. C. Penner

MSC:

32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
14H15 Families, moduli of curves (analytic)
53A35 Non-Euclidean differential geometry
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[1] Abikoff, W.: The real-analytic theory of Teichmüller space. Lecture Notes in Mathematics, Vol. 820. Berlin, Heidelberg, New York: Springer 1980 · Zbl 0452.32015
[2] Bowditch, B., Epstein, D.B.A.: Triangulations associated with punctured surfaces. Topology (1987)
[3] Cassels, J.W.S.: Rational quadratic forms. New York: Academic Press 1978 · Zbl 0395.10029
[4] Epstein, D.B.A., Penner, R.C.: Euclidean decompositions of non-compact hyperbolic manifolds. J. Diff. Geom. (1987) · Zbl 0611.53036
[5] Friedan, D., Shenker, S.H.: The analytic geometry of two dimensional conformal field theory. Nucl. Phys. B. (1987)
[6] Harer, J.: The virtual cohomological dimension of the mapping class group of an oriented surface. Invent. Math84, 157-176 (1986) · Zbl 0592.57009 · doi:10.1007/BF01388737
[7] Harer, J., Zagier, D.: The Euler characteristic of the moduli space of curves. Invent. Math.85, 457-485 (1986) · Zbl 0616.14017 · doi:10.1007/BF01390325
[8] Mosher, L.: Pseudo-anosovs on punctured surfaces. Princeton University thesis (1983)
[9] Penner, R.C.: Perturbative series and the moduli space of Riemann surfaces. J. Diff. Geom. (1987) · Zbl 0669.32013
[10] Penner, R.C.: The moduli space of punctured surfaces. Proceedings of the Mathematical Aspects of String Theory Conference, University of California, San Diego, World Science Press, 1987 · Zbl 0669.32013
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