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


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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