Transversals to horocycle flow on the moduli space of translation surfaces

Work, Grace M

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https://hdl.handle.net/2142/90599 Copy

Description

Title

Transversals to horocycle flow on the moduli space of translation surfaces

Author(s)

Work, Grace M

Issue Date

2016-04-21

Director of Research (if dissertation) or Advisor (if thesis)

Athreya, Jayadev

Doctoral Committee Chair(s)

Kapovich, Ilya

Committee Member(s)

• Dunfield, Nathan
• Tyson, Jeremy

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Translation surfaces
• gap distribution
• saddle connection
• Veech surface
• horocycle flow

Abstract

Computing the distribution of the gaps between slopes of saddle connections is a question that was studied first by Athreya and Cheung in the case of the torus, motivated by the connection with Farey fractions, and then in the case of the golden L by Athreya, Chaika, and Lelievre. Their strategy involved translating the question of gaps between slopes of saddle connections into return times under horocycle flow on the space of translation surfaces to a specific transversal. We show how to use this strategy to explicitly compute the distribution in the case of the octagon, the first case where the Veech group has multiple cusps, how to generalize the construction of the transversal to the general Veech case (both joint work with Caglar Uyanik), and how to parametrize the transversal in the case of a generic surface in H(2).

Graduation Semester

2016-05

Type of Resource

text

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Copyright and License Information

Copyright 2016 Grace Work

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