Examples for Veech groups of origamis

Schmithüsen, Gabriela

doi:10.1090/conm/397/07473

Cited by 23 publications

(29 citation statements)

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“…The proposition is a generalisation of Theorem 1 in [Sch05], using [Sch08]. As stated above, every affine map on¯defines an automorphism ∈ Aut( 1 ( )), well-defined up to an inner automorphism of 1 ( ) ≤ 1 ( ) ∼ = .…”

Section: Definitions and Preliminariesmentioning

confidence: 80%

“…Following the proof of Lemma 6.5 in [Sch05], we now prove a characterisation of paths winding (several times) around a singularity and lying in , := ⟨ , ⟩.…”

Section: Asmentioning

confidence: 98%

“…We address the question, which subgroups of the Veech group of a primitive translation surfacē can be realised as Veech group of a covering surface¯. Our starting point is the work of Gabriela Weitze-Schmithüsen in [Sch05]. There she proves that many congruence subgroups of SL 2 (Z) can be realised as Veech groups of Origamis, i.e.…”

Section: Introductionmentioning

confidence: 99%

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Congruence Veech groups

Finster

2016

Isr. J. Math.

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We study Veech groups of covering surfaces of primitive translation surfaces. Therefore we define congruence subgroups in Veech groups of primitive translation surfaces using their action on the homology with entries in Z/ Z. We introduce a congruence level definition and a property of a primitive translation surface which we call property (⋆). It guarantees that partition stabilising congruence subgroups of this level occur as Veech group of a translation covering.Each primitive surface with exactly one singular point has property (⋆) in every level. We additionally show that the surface glued from a regular 2 -gon with odd has property (⋆) in level iff and are coprime. For the primitive translation surface glued from two regular -gons, where is an odd number, we introduce a generalised Wohlfahrt level of subgroups in its Veech group. We determine the relationship between this Wohlfahrt level and the congruence level of a congruence group.

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Section: Definitions and Preliminariesmentioning

confidence: 80%

“…Following the proof of Lemma 6.5 in [Sch05], we now prove a characterisation of paths winding (several times) around a singularity and lying in , := ⟨ , ⟩.…”

Section: Asmentioning

confidence: 98%

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Congruence Veech groups

Finster

2016

Isr. J. Math.

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“…They consist of the derivatives of affine diffeomorphisms of a finite number of glued copies of the unit square (see e.g. [WS05]). Their study goes back to the fundamental paper [Vee89], where Veech groups have been introduced first.…”

Section: Introductionmentioning

confidence: 99%

Non-congruence of homology Veech groups in genus two

Weiß

2018

Geom Dedicata

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We study the action of the Veech group of square-tiled surfaces of genus two on homology. This action defines the homology Veech group which is a subgroup of SL 2 (O D ) where O D is a quadratic order of square discriminant. Extending a result of Weitze-Schmithüsen we show that also the homology Veech group is a totally non-congruence subgroup with exceptions stemming only from the prime ideals lying above 2. While Weitze-Schmithüsen's result for Veech groups is asymmetric with respect to the spin structure our use of the homology Veech group yields a completely symmetric picture.

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“…Lochak proposed in [23] a combinatorial construction for such coverings (which led to the name "origami"), and Schmithüsen [30] gave a group theoretic characterization of the Veech group. In [31], origamis and their Veech groups are systematically studied and numerous examples are presented. Origamis in genus 2 where q has one zero are classified in [19].…”

Section: Introductionmentioning

confidence: 99%

On the boundary of Teichmüller disks in Teichmüller and in Schottky space

Herrlich

Schmithüsen

2007

IRMA Lectures in Mathematics and Theoretical Physics

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We study the boundary of Teichmüller disks in T g , a partial compactification of Teichmüller space, and their image in Schottky space. We give a broad introduction to Teichmüller disks and explain the relation between Teichmüller curves and Veech groups. Furthermore, we describe Braungardt's construction of T g and compare it with the Abikoff augmented Teichmüller space. Following Masur, we give a description of Strebel rays that makes it easy to understand their end points on the boundary of T g . This prepares the description of boundary points that a Teichmüller disk has, with a particular emphasis to the case that it leads to a Teichmüller curve. Further on we turn to Schottky space and describe two different approaches to obtain a partial compactification. We give an overview how the boundaries of Schottky space, Teichmüller space and moduli space match together and how the actions of the diverse groups on them are linked. Finally we consider the image of Teichmüller disks in Schottky space and show that one can choose the projection from Teichmüller space to Schottky space in such a manner that the image of the Teichmüller disk is a quotient by an infinite group.

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Examples for Veech groups of origamis

Cited by 23 publications

References 0 publications

Congruence Veech groups

Congruence Veech groups

Non-congruence of homology Veech groups in genus two

On the boundary of Teichmüller disks in Teichmüller and in Schottky space

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