Characterizations of lattice surfaces

John Smillie; Barak Weiss

doi:10.1007/s00222-010-0236-0

Characterizations of lattice surfaces

John Smillie, Barak Weiss

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

39 Scopus citations

Abstract

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.

Original language	English
Pages (from-to)	535-557
Number of pages	23
Journal	Inventiones Mathematicae
Volume	180
Issue number	3
DOIs	https://doi.org/10.1007/s00222-010-0236-0
State	Published - 1 Jun 2010

ASJC Scopus subject areas

General Mathematics

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10.1007/s00222-010-0236-0

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abstract = "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.",

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AB - 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.

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Characterizations of lattice surfaces

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