Infinitely generated Veech groups
type: Item
instance of: scholarly article
title: Infinitely generated Veech groups
zbMATH Open document ID: 1056.30044
DOI: 10.1215/S0012-7094-04-12312-8
author: Pascal Hubert
author: Thomas A. Schmidt
published in: Duke Mathematical Journal
publication date: 2004-11-11T00:00:00Z
review text: Veech groups are subgroups of finite index in a Teichmüller modular group. They are subgroups of \(SL(2,\mathbb R)\). The construction is due to Thurston who investigated surface diffeomorphisms by way of locally flat structures. A {translation surface} is a Riemann surface, where, apart from finite set of points ({singularities}), the transition functions are translations. The first main theorem of this paper is that there are translation surfaces whose Veech group is not finitely generated but is Fuchsian of the first kind. This answers a question posed by W. Veech. The paper is devoted more generally to a detailled analysis of the geometry and group theory of Veech groups.
Mathematics Subject Classification ID: 30F30
Mathematics Subject Classification ID: 32G15
zbMATH DE Number: 2114445
zbMATH Keywords: Veech groups
zbMATH Keywords: translation surfaces
zbMATH Keywords: Teichmüller space
zbMATH Keywords: holonomy field
zbMATH Keywords: affine diffeomorphisms
reviewed by: Samuel James Patterson
MaRDI profile type: Publication
cites work: Continued fractions on the Veech surfaces
cites work: Q3310101
cites work: Singular Euclidean Structures on Surfaces
cites work: Q4720067
cites work: Veech surfaces and complete periodicity in genus two
cites work: Q5284127
cites work: Q4002564
cites work: Billiards on almost integrable polyhedral surfaces
cites work: Affine diffeomorphisms of translation surfaces: Periodic points, Fuchsian groups, and arithmeticity
cites work: The geometry and arithmetic of translation surfaces with applications to polygonal billiards
cites work: Affine mappings of translation surfaces: Geometry and arithmetic
cites work: Polygonal billiards with small obstacles
cites work: Veech groups and polygonal coverings
cites work: Invariants of translation surfaces
cites work: Q4004260
cites work: Q4076561
cites work: Billiards on rational-angled triangles
cites work: Ergodicity of billiard flows and quadratic differentials
cites work: Connected components of the moduli spaces of Abelian differentials with prescribed singularities
cites work: On the Nielsen-Thurston-Bers type of some self-maps of Riemann surfaces
cites work: Über die Heckeschen Gruppen \({\mathfrak G}(\lambda)\)
cites work: Über die Heckeschen Gruppen \(G(\lambda)\). II
cites work: Hausdorff dimension of the set of nonergodic foliations of a quadratic differential
cites work: Hausdorff dimension of sets of nonergodic measured foliations
cites work: Q4474941
cites work: Billiards and Teichmüller curves on Hilbert modular surfaces
cites work: Teichmüller geodesics of infinite complexity.
cites work: On triangular billiards
cites work: On the geometry and dynamics of diffeomorphisms of surfaces
cites work: Q4337982
cites work: Q4231188
cites work: Les surfaces euclidiennes à singularités coniques. (Euclidean surfaces with cone singularities)
cites work: Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards
cites work: The billiard in a regular polygon
cites work: Q4891867
cites work: Measures supported on the set of uniquely ergodic directions of an arbitrary holomorphic 1-form
cites work: Planar structures and billiards in rational polygons: the Veech alternative
cites work: Calculation of Fuchsian groups associated to billiards in a rational triangle
cites work: Eine Bemerkung über Heckes Modulgruppen
full work available at URL: s0012-7094-04-12312-8
OpenAlex ID: W1968754113
description: scientific article; zbMATH DE number 2114445