Full rank affine invariant submanifolds

TL;DR

This paper classifies GL(2, R) orbit closures of translation surfaces, revealing their structure and implications for polygonal billiards, including the existence of infinitely many rational triangles with dense orbits.

Contribution

It proves a classification theorem for GL(2, R) orbit closures, identifying their types and applications to polygonal billiards.

Findings

Orbit closures are either connected components of strata, hyperelliptic loci, or surfaces with extra Jacobian endomorphisms.

Existence of infinitely many rational triangles with unfoldings having dense GL(2,R) orbits.

Applications to understanding billiard dynamics in rational polygons.

Abstract

We show that every GL(2, R) orbit closure of translation surfaces is either a connected component of a stratum, the hyperelliptic locus, or consists entirely of surfaces whose Jacobians have extra endomorphisms. We use this result to give applications related to polygonal billiards. For example, we exhibit infinitely many rational triangles whose unfoldings have dense GL(2,R) orbit.