Teichmüller rays and the Gardiner-Masur boundary of Teichmüller space
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.
scientific article; zbMATH DE number 5373526
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