An Introduction to Veech Surfaces

Hubert, Pascal; Schmidt, Thomas A.

doi:10.1016/s1874-575x(06)80031-7

Cited by 76 publications

(59 citation statements)

References 24 publications

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“…We refer the reader to [6,10] for more details on Veech groups of compact flat surfaces, and to [3,4,9,2] for explicit examples of Veech groups of tame flat surfaces which are Loch Ness monsters.…”

Section: Preliminariesmentioning

confidence: 99%

“…Thus Theorem 1.2 implies that every discrete subgroup of SL(2, R) (in particular, every cyclic subgroup of SL(2, R) and the pre-image in SL(2, R) of every Fuchsian group) can be realized as the Veech group of a tame flat surface which is a Loch Ness monster. For compact flat surfaces, such questions are still open (see [5,Problems 5,6]). Furthermore, observe that the pre-image in SL(2, R) of a cocompact Fuchsian group cannot be the Veech group of a compact flat surface [10], but it occurs as the Veech group of a tame flat surface, which is a Loch Ness monster.…”

Section: Introductionmentioning

confidence: 99%

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Veech Groups of Loch Ness Monsters

Przytycki¹,

Schmithüsen²,

Valdez³

2011

Ann. inst. Fourier

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We classify Veech groups of tame non-compact flat surfaces. In particular we prove that all countable subgroups of GL + (2, R) avoiding the set of mappings of norm less than 1 appear as Veech groups of tame non-compact flat surfaces which are Loch Ness monsters. Conversely, a Veech group of any tame flat surface is either countable, or one of three specific types. * Partially supported by MNiSW grant N201 012 32/0718 and the Foundation for Polish Science.† Partially supported by Landesstiftung Baden-Württemberg.‡ Partially supported by Sonderforschungsbereich/Transregio 45 and ANR Symplexe.

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Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Veech Groups of Loch Ness Monsters

Przytycki¹,

Schmithüsen²,

Valdez³

2011

Ann. inst. Fourier

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“…For more on translation surfaces, see the introductory texts [Es06], [Fo06], [HS06], [Ma86], [MaTa02], [Vi07], [Yo06], [Zo06].…”

Section: + 2 (R)-actionmentioning

confidence: 99%

“…The Veech group of a generic surface is trivial. However, Veech groups can be complicated objects (see [HS06], [Mc2] for instance). Closed GL + 2 (R)-orbits can be characterized in terms of Veech groups.…”

Section: Veech Groups and Veech Surfaces Closed Glmentioning

confidence: 99%

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GL⁺₂(R)-orbit closures via topological splittings

Hubert

Lanneau

Möller

2009

Surveys in Differential Geometry

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A Short Introduction to Translation Surfaces, Veech Surfaces, and Teichmüller Dynamics

Massart

2022

Surveys in Geometry I

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We review the different notions about translation surfaces which are necessary to understand McMullen's classification of GL + 2 (R)-orbit closures in genus two. In Section 2 we recall the different definitions of a translation surface, in increasing order of abstraction, starting with cutting and pasting plane polygons, ending with Abelian differentials. In Section 3 we define the moduli space of translation surfaces and explain its stratification by the type of zeroes of the Abelian differential, the local coordinates given by the relative periods, its relationship with the moduli space of complex structures and the Teichműller geodesic flow. In Part II we introduce the GL + 2 (R)-action, and define the related notions of Veech group, Teichműller disk, and Veech surface. In Section 8 we explain how McMullen classifies GL + 2 (R)-orbit closures in genus 2: you have orbit closures of dimension 1 (Veech surfaces, of which a complete list is given), 2 (Hilbert modular surfaces, of which again a complete list is given), and 3 (the whole moduli space of complex structures). In the last section we review some recent progress in higher genus.

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An Introduction to Veech Surfaces

Cited by 76 publications

References 24 publications

Veech Groups of Loch Ness Monsters

Veech Groups of Loch Ness Monsters

GL⁺₂(R)-orbit closures via topological splittings

A Short Introduction to Translation Surfaces, Veech Surfaces, and Teichmüller Dynamics

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