Finiteness of Teichm\"uller curves in non-arithmetic rank 1 orbit closures

TL;DR

This paper proves that non-arithmetic rank 1 orbit closures of translation surfaces contain finitely many Teichmüller curves and characterizes completely parabolic surfaces as Veech surfaces within these closures.

Contribution

It establishes finiteness of Teichmüller curves in non-arithmetic rank 1 orbit closures and identifies completely parabolic surfaces as Veech surfaces in this setting.

Findings

Finitely many Teichmüller curves in non-arithmetic rank 1 orbit closures.

Completely parabolic surfaces are Veech surfaces in these closures.

Provides structural insights into the geometry of translation surfaces.

Abstract

We show that in any non-arithmetic rank 1 orbit closure of translation surfaces, there are only finitely many Teichm\"uller curves. We also show that in any non-arithmetic rank 1 orbit closure, any completely parabolic surface is Veech.