Convergence of some horocyclic deformations to the Gardiner-Masur boundary (Q2790832)

The author introduces a \textit{horocyclic deformation} of Riemann surfaces, which is essentially an embedding of the real line in the Teichmüller space \(\mathcal{T}(X_0)\), whose image coincides with a certain horocycle in a Teichmüller disk.NEWLINENEWLINELet \(F\) be a measured foliation on \(X_0\), and let \(x=(X,f)\) be a point in the Teichmüller space \(\mathcal{T}(X_0)\). We denote by \(\mathcal{R}^t_{[F]}\) the Teichmüller deformation directed by \(F\), and by \(\mathcal{H}^s_{[F]}\) the horocyclic deformation directed by \(F\), where \(s,t\) are both real numbers. We refer to Section 3 of the paper for the definitions.NEWLINENEWLINEThe author shows some basic properties of the horocyclic deformation (Section 3.3):NEWLINENEWLINE1) There is a horocyclic deformation between any two points of the Teichmüller space. NEWLINENEWLINENEWLINENEWLINE 2) The horocyclic deformation and the Teichmüller deformation commute: \(\mathcal{H}^s_{[F]}\circ\mathcal{R}^t_{[F]}=\mathcal{R}^t_{[F]}\circ\mathcal{H}^s_{[F]}\) for any real \(s,t\). NEWLINENEWLINENEWLINENEWLINE 3) The horocyclic deformation perserves the extremal length of the measured foliation: \( \text{Ext}_{\mathcal{H}^s_{[F]}(x)}(F)=\text{Ext}_x(F)\) for any \(s\), where \(\text{Ext}_x(F)\) is the extremal length of \(F\) on \(X\). NEWLINENEWLINENEWLINENEWLINE The Teichmüller deformation and the horocyclic deformation use the extremal length, which naturally appears when \(\mathcal{T}(X_0)\) is defined from the conformal point of view. An alternative definition of the Teichmüller space uses the hyperbolic metric, which leads to two other deformations: the earthquakes and the stretches. The above-stated properties can be compared to those of the earthquakes and stretches from the hyperbolic standpoint [\textit{S. P. Kerckhoff}, Ann. Math. (2) 117, 235--265 (1983; Zbl 0528.57008); \textit{G. Théret}, A propos de la métrique asymétrique de Thurston sur l'espace de Teichmüller d'une surface. Strasbourg: Université Louis Pasteur (PhD thesis) (2005)]. In particular, the horocyclic deformation can be seen as an analogue of the earthquake deformation. The two perspectives of the Teichmüller space lead to two compactifications: the Gardiner-Masur compactification (conformal) and the Thurston compactification (hyperbolic).NEWLINENEWLINEThe main theorem of the paper shows that the horocyclic deformation converges in the Gardiner-Masur compactification in two particular cases: \(F\) is either a uniquely ergodic measured foliation, or \(F\) is a simple closed curve. Moreover, the limits can be computed, and they are seen to be equal to the limit of the corresponding Teichmüller deformation in the Gardiner-Masur compactification. Namely, the author shows that, in the above-mentioned two cases, NEWLINE\[NEWLINE \lim_{t\to\pm\infty}\mathcal{H}^t_{[F]}= [F] =\lim_{t\to\pm\infty}\mathcal{R}^t_{[F]}. NEWLINE\]NEWLINE One can also see that in these two cases, the limits of the horocylic deformation are also contained in the Thurston compactification.NEWLINENEWLINEThe author also gives an example of a horocylic deformation that converges to a limit which is different from the limit of the corresponding Teichmüller deformation.