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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
##### Keywords:
decorated Teichmüller space; ideal cell decomposition
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##### References:
 [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 [7] Harer, J., Zagier, D.: The Euler characteristic of the moduli space of curves. Invent. Math.85, 457-485 (1986) · Zbl 0616.14017 [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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