REVIEW OF: "Rafi Kasra - Schleimer Saul, Curve complexes are rigid, Duke Math. J. 158, No. 2, 225-246 (2011)".[DE059178142]

The {\it curve complex} of a surface was introduced into the study of Teichmüller space by Harvey (see [Riemann surfaces and related topics: Proc. 1978 Stony Brook Conf., Ann. Math. Stud. 97, 245-251 (1981; Zbl 0461.30036)]) as an analogous of the Tits building of a symmetric space. The present paper deals about geometric structure of the curve complex (see [Invent. Math. 138, No.1, 103-149 (1999; Zbl 0941.32012)] and [Geom. Funct. Anal. 10, No.4, 902-974 (2000; Zbl 0972.32011)]) of an orientable connected compact surface $S$, in case the {\it complexity} $\csi(S)$ of $S$ is greater or equal to two, where $\csi(S)=3g-3+b$, $g$ (resp. $b$) being the genus (resp. the number of boundary components) of $S$. By making use of the notions of {\it cobounded ending lamination} and of {\it marking complex}, together with some key results due to Gabai (see [Geom. Topol. 13, No. 2, 1017-1041 (2009; Zbl 1165.57015)]) and to Berhrstock-Kleiner-Minsky-Mosher (see [{\it Geometry and rigidity of mapping class group}, arXiv:0801.2006v4]), the authors prove that any quasi isometry of the curve complex is bounded distance from a simplicial automorphism. As a consequence, the quasi-isometry type of the curve complex determines the homeomorphism type of the surface