Finiteness results for flat surfaces: large cusps and short geodesics

Smillie, John; Weiss, Barak

doi:10.4171/cmh/197

Cited by 4 publications

(2 citation statements)

References 20 publications

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“…However, if one considers any fixed stratum, the number of square-tiled surfaces with a fixed isomorphism class of the Veech group is finite within this stratum (see [SW,Corollary 1.7]). Nevertheless, these square-tiled surfaces might be distributed into several PSL(2, Z)orbits (see, say, Example 5.3 of F. Nisbach's Ph.D. thesis [N]).…”

Section: Closed Geodesics On An Arithmetic Teichmüller Curvementioning

confidence: 99%

Zero Lyapunov exponents of the Hodge bundle

Forni¹,

Matheus²,

Zorich³

2014

Comment. Math. Helv.

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ABSTRACT. By the results of G. Forni and of R. Treviño, the Lyapunov spectrum of the Hodge bundle over the Teichmüller geodesic flow on the strata of Abelian and of quadratic differentials does not contain zeroes even though for certain invariant submanifolds zero exponents are present in the Lyapunov spectrum. In all previously known examples, the zero exponents correspond to those PSL(2, R)-invariant subbundles of the real Hodge bundle for which the monodromy of the Gauss-Manin connection acts by isometries of the Hodge metric. We present an example of an arithmetic Teichmüller curve, for which the real Hodge bundle does not contain any PSL(2, R)-invariant, subbundles, and nevertheless its spectrum of Lyapunov exponents contains zeroes. We describe the mechanism of this phenomenon; it covers the previously known situation as a particular case. Conjecturally, this is the only way zero exponents can appear in the Lyapunov spectrum of the Hodge bundle for any PSL(2, R)-invariant probability measure.

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Section: Closed Geodesics On An Arithmetic Teichmüller Curvementioning

confidence: 99%

Zero Lyapunov exponents of the Hodge bundle

Forni¹,

Matheus²,

Zorich³

2014

Comment. Math. Helv.

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“…Understanding Veech groups is a central problem in Teichmüller dynamics. Various aspects of this problem has been addressed by several authors, see for instance [15,16,13,21,26,27,39,40,2,36,45,33]. The goal of this paper is to contribute to the investigation of Veech groups by using the properties of the flat metric, especially for surfaces satisfying the topological Veech dichotomy.…”

Section: 5])mentioning

confidence: 99%

Topological Veech dichotomy and tessellations of the hyperbolic plane

Nguyen¹

2018

Preprint

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We construct for every half-translation surface satisfying the topological Veech dichotomy a tessellation of the Poincaré upper half plane. This construction can be viewed as a generalization of the Farey tessellation for a flat torus. As a consequence, we get a bound on the volume of the corresponding Teichmüller curve for a lattice surface (Veech surface). There is a natural graph underlying this tessellation on which the affine group acts by automorphisms. We provide algorithms to determine a "coarse" fundamental domain and a generating set for the Veech group based on this graph. We also show that this graph has infinite diameter and is Gromov hyperbolic.

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Constructing lattice surfaces with prescribed Veech groups: an algorithm

Sanderson

2023

Geom Dedicata

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The Veech group of a translation surface is the group of Jacobians of orientation-preserving affine automorphisms of the surface. We present an algorithm which constructs all translation surfaces with a given lattice Veech group in any given stratum. In developing this algorithm, we give a new proof of a finiteness result of Smillie and Weiss, namely that there are only finitely many unit-area translation surfaces in any stratum with the same lattice Veech group. Our methods can be applied to obtain obstructions of lattices being realized as Veech groups in certain strata; in particular, we show that the square torus is the only translation surface in any minimal stratum whose Veech group is all of $$\textrm{SL}_2\mathbb {Z}$$ SL 2 Z .

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Finiteness results for flat surfaces: large cusps and short geodesics

Cited by 4 publications

References 20 publications

Zero Lyapunov exponents of the Hodge bundle

Zero Lyapunov exponents of the Hodge bundle

Topological Veech dichotomy and tessellations of the hyperbolic plane

Constructing lattice surfaces with prescribed Veech groups: an algorithm

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