A Thurston boundary and visual sphere of the universal Teichmüller space
type: Item
instance of: scholarly article
title: A Thurston boundary and visual sphere of the universal Teichmüller space
zbMATH Open document ID: 1482.30112
author: Hrant Hakobyan
author: Dragomir Šarić
published in: Journal d'Analyse Mathématique
publication date: 2021-08-17T00:00:00Z
review text: The universal Teichmüller space \(\mathcal{T}(\mathbb{D})\) is the space of all quasisymmetric homeomorphisms of \(\partial \mathbb{D}=\mathbb{S}^1\) subject to a normalization condition (i.e., fixing 1, \(i\) and \(-1\) in \(\mathbb{S}^1\)). The universal Teichmüller space homeomorphically embeds in bounded geodesic current space as Liouville geodesic currents [\textit{F. Bonahon} and the second author, Math. Ann. 380, No. 3--4, 1119--1167 (2021; Zbl 1472.57016)], generalizing Bonahon's geodesic current theory for closed surfaces [\textit{F. Bonahon}, Invent. Math. 92, No. 1, 139--162 (1988; Zbl 0653.32022)]. The second author [Topology 44, No. 1, 99--130 (2005; Zbl 1082.30036)], and \textit{F. Bonahon} and the second author [Math. Ann. 380, No. 3--4, 1119--1167 (2021; Zbl 1472.57016)] used this setup to define a natural generalization of the Thurston bordification of Teichmüller space in terms of adding bounded projective measured laminations \(\mathcal{PML}_{bdd}(\mathbb{D})\) -- a space consisting of \(\mathbb{R}_+\)-rays of \textit{bounded geodesic currents} asymptotic to the embedding of \(\mathcal{T}(\mathbb{D})\). The primary aim of this paper is to establish the convergence, to \(\mathcal{PML}_{bdd}(\mathbb{D})\), of certain families of of Teichmüller metric geodesics in \(\mathcal{T}(\mathbb{D})\).\medskip \textit{H. Masur} [Duke Math. J. 49, 183--190 (1982; Zbl 0508.30039)] showed that for Teichmüller spaces \(\mathcal{T}(S)\) of closed surfaces \(S\), Teichmüller metric geodesic rays, obtained from shrinking the vertical foliations for integrable holomorphic quadratic differentials whose vertical foliations are either \begin{itemize} \item[1.] uniquely ergodic, or \item[2.] consist of finitely many cylinders, \end{itemize} necessarily converge to a single point on the Thurston boundary (i.e., the projective measured lamination space \(\mathcal{PML}(S)\) of \(\mathcal{T}(S)\)). Case~1 converges to the unique projective measured lamination corresponding to…
reviewed by: Yi Huang
Mathematics Subject Classification ID: 30F60
Mathematics Subject Classification ID: 32G15
zbMATH DE Number: 7382547
zbMATH Keywords: universal Teichmüller space
zbMATH Keywords: quasisymmetric homeomorphisms of \(\mathbb{S}^1\)
zbMATH Keywords: geodesics
MaRDI profile type: Publication
full work available at URL: s11854-021-0166-3
OpenAlex ID: W3174552718
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DOI: 10.1007/S11854-021-0166-3
description: scientific article; zbMATH DE number 7382547