Visual sphere and Thurston's boundary of the Universal Teichm\"uller space

TL;DR

This paper explores the boundary behavior of the universal Teichmüller space, focusing on geodesic rays induced by both integrable and non-integrable holomorphic quadratic differentials, revealing complex limit lamination structures.

Contribution

It extends the understanding of Thurston's boundary by analyzing limits of geodesic rays from non-integrable quadratic differentials, showing unexpected lamination support structures.

Findings

Limits of geodesic rays can have support not homotopic to vertical or horizontal foliations.

Non-integrable differentials induce geodesic rays with complex boundary limits.

Supports of limit laminations may be disconnected or non-homotopic to known foliations.

Abstract

Thurston's boundary to the universal Teichm\"uller space $T(\mathbb{D})$ is the space $PML_{bdd}(\mathbb{D})$ of projective bounded measured laminations of $\mathbb{D}$. A geodesic ray in $T(\mathbb{D})$ is of Teichm\"uller type if it shrinks vertical foliation of an integrable holomorphic quadratic differential. In a prior work we established that each Teichm\"uller geodesic ray limits to a multiple (by the reciprocal of the length of the leaves) of vertical foliation of the quadratic differential. Certain non-integrable holomorphic quadratic differential induce geodesic rays and we consider their limit points in $PML_{bdd}(\mathbb{D})$. Somewhat surprisingly, the support of the limiting projective measured laminations might be a geodesic lamination whose leaves are not homotopic to leaves of either vertical or horizontal foliation of the non-integrable holomorphic quadratic differential.