Transversals to horocycle flow on the moduli space of translation surfaces

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).

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2016

Author dc:creator

• Work, Grace M

Contributors dc:contributor

• Athreya, Jayadev
• Kapovich, Ilya
• Dunfield, Nathan
• Tyson, Jeremy

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

Statement dc:rights

• Copyright 2016 Grace Work

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/90599

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/90599