Visual sphere and Thurston's boundary of the Universal Teichm\"uller space
Hrant Hakobyan, Dragomir Saric

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 is the space of projective bounded measured laminations of . A geodesic ray in 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 . 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…
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Taxonomy
TopicsGeometric and Algebraic Topology · Geometric Analysis and Curvature Flows · Point processes and geometric inequalities
