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Slope gap distributions of Veech surfaces

Luis Kumanduri, Anthony Sanchez and Jane Wang

Algebraic & Geometric Topology 24 (2024) 951–980

The slope gap distribution of a translation surface is a measure of how random the directions of the saddle connections on the surface are. It is known that Veech surfaces, a highly symmetric type of translation surface, have gap distributions that are piecewise real analytic. Beyond that, however, very little is currently known about the general behavior of the slope gap distribution, including the number of points of nonanalyticity or the tail.

We show that the limiting gap distribution of slopes of saddle connections on a Veech translation surface is always piecewise real analytic with finitely many points of nonanalyticity. We do so by taking an explicit parametrization of a Poincaré section to the horocycle flow on SL (2, )SL (X,ω) associated to an arbitrary Veech surface (X,ω), and establishing a key finiteness result for the first return map under this flow. We use the finiteness result to show that the tail of the slope gap distribution of a Veech surface always has quadratic decay.

gap distributions, slope gaps, saddle connections, holonomy vectors, translation surfaces, Veech surfaces
Mathematical Subject Classification
Primary: 32G15, 37D40
Secondary: 14H55
Received: 5 December 2021
Revised: 5 August 2022
Accepted: 3 September 2022
Published: 12 April 2024
Luis Kumanduri
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA
United States
Anthony Sanchez
Department of Mathematics
University of California San Diego
La Jolla, CA
United States
Jane Wang
Department of Mathematics and Statistics
University of Maine
Orono, ME
United States

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