Isometries of Teichmüller spaces from the point of view of Mostow rigidity. (Q2735116)

Let \(S\) be a surface which is neither a sphere with \(\leq 4\) punctures nor a torus with \(\leq 2\) punctures. Let \(T_S\) be the Teichmüller space of \(S\) equipped with the Teichmüller metric, and let \(Mod_S\) be the mapping class group of \(S\). Finally, let \(C(S)\) be the \textit{complex of curves of \(S\)}. This is the simplicial complex whose vertices are the isotopy classes of nontrivial simple closed curves on \(S\) and where a set of vertices is declared to be a simplex if and only if these vertices can be represented by a set of pairwise disjoint closed curves on \(S\). In this paper, the author gives a new proof of the following theorem, which is due to Royden in the case where \(S\) closed and to Earle-Kra in the case where \(S\) has punctures. NEWLINENEWLINETheorem A: All isometries of \(T_S\) belong the \(Mod_S\). NEWLINENEWLINEThe author deduces Theorem A from the following NEWLINENEWLINETheorem B: All automorphisms of \(C(S)\) are given by elements of \(Mod(S)\). NEWLINENEWLINETheorem B has been proved by the author in previous work, in the case where \(S\) has genus \(\geq 2\), and by Korkmaz in the remaining case. The author deduces Theorem B from Theorem A in a manner which is inspired from Mostow's proof of his rigidity theorem. Here, the author makes use of Thurston's boundary of Teichmüller space, which is the space of projective measured foliations. For that purpose, he intorduces notions of \textit{divergent} and of \textit{parallel} rays in Teichmüller space. He develops sufficient conditions for rays to be divergent, and for rays to be parallel. Besides their use in the proof of Theorem A, these results have their own interest.NEWLINENEWLINEFor the entire collection see [Zbl 0961.00011].