Cylinder deformations in orbit closures of translation surfaces

By A. Wright. 2013

99 citations recorded by the source corpus.

Let M be a translation surface. We show that certain deformations of M supported on the set of all cylinders in a given direction remain in the GL(2,R)-orbit closure of M. Applications are given concerning complete periodicity, field of definition, and the number of of parallel cylinders which may be found on a translation surface in a given orbit closure. The proof uses Eskin-Mirzakhani-Mohammadi's recent theorem on orbit closures of translation surfaces, as well as results of Minsky-Weiss and Smillie-Weiss on the dynamics of horocycle flow.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e04de26306a1a5da5fc13e6cd2f8ccd542716eec

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In Geometry and Topology

1. Dawei Chen
2. A. Wright
3. Don Zagier
4. Simion Filip
5. A. Eskin
6. Martin Möller
7. A. Sauvaget
8. Maryam Mirzakhani

In Mathematical Physics

1. Dawei Chen
2. Don Zagier
3. A. Wright
4. Martin Möller
5. A. Sauvaget
6. A. Eskin
7. Simion Filip
8. Maryam Mirzakhani

Citations in the corpus

• A tour through Mirzakhani’s work on moduli spaces of Riemann surfaces (2019, methodology): “W17], see also [CW]. This was later used in [MW18] to prove the conjecture in the case of maximum rank, which is rank equal to the genus. This also used the author’s Cylinder Deformation Theorem from [Wri15a], which says that often cylinders in translation surfaces can be deformed while staying in t”
• The algebraic hull of the Kontsevich--Zorich cocycle (2017, background): “This particular consequence of [Fil17] can also be derived as follows, assuming familiarity with [Wri15a].”
• Masur–Veech volumes and intersection theory on moduli spaces of Abelian differentials (2019, methodology): “ctions on (X,ω) ∈ M are M-parallel if they are parallel and stay parallel in a neighborhood of (X,ω) in M. (The terminology is completely analogous to the notion of M-parallel cylinders introduced in [Wri15].) We thus define the counting function AM−p 1↔2 (T) and the Siegel-Veech constant cM−p 1↔2 (M”