Teichmüller rays and the Gardiner-Masur boundary of Teichmüller space
type: Item
instance of: scholarly article
title: Teichmüller rays and the Gardiner-Masur boundary of Teichmüller space
zbMATH Open document ID: 1165.30023
author: Hideki Miyachi
published in: Geometriae Dedicata
publication date: 2008-11-25T00:00:00Z
review text: In [Complex Variables, Theory Appl. 16, No. 2--3, 209--237 (1991; Zbl 0702.32019)], \textit{F. Gardiner} and \textit{H. Masur} introduced a new compactification of the Teichmüller space \(\mathcal{T}(S)\) of a compact oriented surface \(S\) of genus \(g\) by mimicking Thurston's idea, but replacing the hyperbolic length by the extremal length. The result is that Gardiner-Masur's points at infinity in \(\overline{\mathcal{T}}^{GM}(S)\) record the projective class of the extremal length spectrum. In the first part of the present paper, the author shows that \(\overline{\mathcal{T}}^{GM}(S)\) is metrizable and that the natural action of the mapping class group \(\mathrm{Mod}(S)\) on \(\mathcal{T}(S)\) extends to \(\overline{\mathcal{T}}^{GM}(S)\). In their paper, Gardiner and Masur had proven that their boundary \(\partial^{GM}\mathcal{T}(S)\) contains Thurs-ton's \(\partial^{Th}\mathcal{T}(S)\). In the second part of the present paper, the author examines the behaviour at infinity of Teichmüller geodesic and he proves that: 1) Teichmüller geodesics can accumulate at a projective class of a rational measured lamination \([\lambda]\) only if the support of \(\lambda\) does not contain two simple closed curves; and 2) the Teichmüller geodesic starting at \(X\in\mathcal{T}(S)\) and associated to a rational measured lamination \(\mu\) limits to \([\lambda]\in\partial^{GM}\mathcal{T}(S)\) if and only if \(\mu\) is supported on a simple closed curve. The situation shows some tension with a result of \textit{S. Kerckhoff} [Topology 19, 23--41 (1980; Zbl 0439.30012)] recalled by the author, which essentially implies that the Gardiner-Masur boundary contains and distinguishes the limit points of Teichmüller geodesics starting at some \(X\in\mathcal{T}(S)\) and associated to a rational measured lamination. In his computations, the authors uses Kerckhoff's extremal length estimates, which are recalled in the appendix.
reviewed by: Gabriele Mondello
Mathematics Subject Classification ID: 30F60
Mathematics Subject Classification ID: 32G15
zbMATH DE Number: 5373526
zbMATH Keywords: Teichmüller space
zbMATH Keywords: Teichmüller geodesics
zbMATH Keywords: extremal length
zbMATH Keywords: Thurston boundary
zbMATH Keywords: measured foliations
zbMATH Keywords: quadratic differentials
MaRDI profile type: Publication
full work available at URL: s10711-008-9289-2
OpenAlex ID: W2068048983
cites work: Q5512337
cites work: Extremal length geometry of teichmüller space
cites work: Quadratic differentials and foliations
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cites work: Two boundaries of Teichmüller space
cites work: Teichmüller geodesics and ends of hyperbolic 3-manifolds
cites work: Q3504693
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DOI: 10.1007/S10711-008-9289-2
description: scientific article; zbMATH DE number 5373526