Broken hyperbolic structures and affine foliations on surfaces.
type: Item
instance of: scholarly article
title: Broken hyperbolic structures and affine foliations on surfaces.
zbMATH Open document ID: 1060.37017
DOI: 10.1016/S0019-3577(04)90019-5
author: Ulrich Oertel
author: Athanase Papadopoulos
published in: Indagationes Mathematicae. New Series
publication date: 2005-01-03T00:00:00Z
review text: Let \(S\) be an oriented surface of negative Euler characteristic with at least one puncture. The authors investigate the following extension of Teichmüller space for \(S\): First, choose an ideal triangulation \( \lambda\) of \(S\), i.e., a decomposition of \(S\) into triangles whose edges are homotopically nontrivial arcs with endpoints at the punctures of \(S\). It is well known that Teichmüller space can be described by glueing coordinates where ideal hyperbolic triangles are glued isometrically along their boundaries according to the combinatorial pattern described by \(\lambda\) and with the additional condition that the glueing weights along all sides going into a fixed puncture add up to zero. Here, the authors allow that the glueing is by homotheties of the edges which are not isometries, with a condition for each cusp corresponding to the weight condition at the punctures for Teichmüller space. In this case, each weight system describing the homotheties defines an element \(\varphi \in H^1 (S; \mathbb R)\). If this element is fixed, then the corresponding system of structures equipped with a suitable topology is shown to be homeomorphic to a ball of dimension \(6g - 6 + 2c\) where \(c > 0\) is the number of punctures. Moreover, it can be compactified by adding a boundary sphere which in turn can be described as a family of ``broken measured foliations'', a generalization of the usual space of measured foliations which compactifies Teichmüller space.
Mathematics Subject Classification ID: 37C85
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57R30
zbMATH DE Number: 2124643
zbMATH Keywords: Teichmüller space for surfaces with punctures
zbMATH Keywords: glueing coordinates
zbMATH Keywords: homothetics
zbMATH Keywords: broken measured foliations
MaRDI profile type: Publication
cites work: Affine foliations and covering hyperbolic structures
cites work: Sur le bord de Thurston de l'espace de Teichmüller d'une surface non compacte. (On the Thurston boundary of the Teichmüller space of a non- compact surface)
cites work: The Weil-Petersson Symplectic Structure at Thurston's Boundary
Wikidata QID: Q125803007
full work available at URL: s0019-3577(04)90019-5
OpenAlex ID: W2049650310
description: scientific article; zbMATH DE number 2124643