Strongly Obtuse Rational Lattice Triangles

TL;DR

This paper classifies certain rational triangles based on their unfolding properties into Veech surfaces, identifying conditions under which they do or do not produce such surfaces, and explores the structure of their orbit closures.

Contribution

It provides a classification of rational triangles with large angles that unfold into Veech surfaces, extending previous work with new criteria and identifying specific infinite families.

Findings

Unfoldings of triangles with largest angle > 2π/3 are generally not Veech surfaces.

Identifies six infinite families where Veech surface unfolding may occur.

Most orbit closures of these unfoldings do not have rank 1.

Abstract

We classify rational triangles which unfold to Veech surfaces when the largest angle is at least $\frac{3\pi}{4}$. When the largest angle is greater than $\frac{2\pi}{3}$, we show that the unfolding is not Veech except possibly if it belongs to one of six infinite families. Our methods include a criterion of Mirzakhani and Wright that built on work of M\"oller and McMullen, and in most cases show that the orbit closure of the unfolding cannot have rank 1.