Characterizations of lattice surfaces

By J. Smillie, Barak Weiss. 2008

77 citations recorded by the source corpus.

We answer a question of Vorobets by showing that the lattice property for flat surfaces is equivalent to the existence of a positive lower bound for the areas of affine triangles. We show that the set of affine equivalence classes of lattice surfaces with a fixed positive lower bound for the areas of triangles is finite and we obtain explicit bounds on its cardinality. We deduce several other characterizations of the lattice property.

Subfields: Geometry and Topology, Materials Chemistry, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/367233c56b372eff06fa775630384842340468af

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Carlos Matheus
2. G. Forni
3. Martin Möller
4. J. Yoccoz
5. C. Ulcigrai
6. J. Smillie
7. A. Zorich
8. A. Eskin

In Materials Chemistry

1. C. Ulcigrai
2. Carlos Matheus
3. J. Smillie

In Mathematical Physics

1. Carlos Matheus
2. Martin Möller
3. G. Forni
4. J. Yoccoz
5. C. Ulcigrai
6. J. Smillie
7. A. Zorich
8. Marcelo Viana

Citations in the corpus

• Introduction to Teichm\"uller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards (2013, background): “Smillie [71], a SL(2,R)-orbit SL(2,R) · (M,ω) is closed if and only if (M,ω) is a Veech surface:”
• Beyond Sturmian sequences: coding linear trajectories in the regular octagon (2011, methodology): “†These permutations are analogous to the ones used to describe glueing patterns in [9] or [20].”
• A criterion for the simplicity of the Lyapunov spectrum of square-tiled surfaces (2013, background): “[39]), the SL2(R)-orbit of (M, ω) is closed in the corresponding moduli space.”