The decorated Teichmüller space of punctured surfaces
type: Item
instance of: scholarly article
title: The decorated Teichmüller space of punctured surfaces
zbMATH Open document ID: 0642.32012
DOI: 10.1007/BF01223515
author: Robert Penner
published in: Communications in Mathematical Physics
publication date: 1987-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 32G15
Mathematics Subject Classification ID: 14H15
Mathematics Subject Classification ID: 53A35
zbMATH DE Number: 4046118
zbMATH Keywords: decorated Teichmüller space
zbMATH Keywords: ideal cell decomposition
MaRDI profile type: Publication
cites work: The real analytic theory of Teichmüller space
cites work: Q4177654
cites work: Euclidean decompositions of noncompact hyperbolic manifolds
cites work: The virtual cohomological dimension of the mapping class group of an orientable surface
cites work: The Euler characteristic of the moduli space of curves
cites work: Q3822475
Wikidata QID: Q125578586
description: scientific article; zbMATH DE number 4046118