TAILIEUCHUNG - Đề tài "Dimension and rank for mapping class groups "

We study the large scale geometry of the mapping class group, MCG(S). Our main result is that for any asymptotic cone of MCG(S), the maximal dimension of locally compact subsets coincides with the maximal rank of free abelian subgroups of MCG(S). An application is a proof of Brock-Farb’s Rank Conjecture which asserts that MCG(S) has quasi-flats of dimension N if and only if it has a rank N free abelian subgroup. (Hamenstadt has also given a proof of this conjecture, using different methods.) We also compute the maximum dimension of quasi-flats in Teichmuller space with the Weil-Petersson metric | Annals of Mathematics Dimension and rank for mapping class groups By Jason A. Behrstock and Yair N. Minsky Annals of Mathematics 167 2008 1055-1077 Dimension and rank for mapping class groups By Jason a. Behrstock and Yair N. MiNSKy Dedicated to the memory of Candida Silveira. Abstract We study the large scale geometry of the mapping class group MCG S . Our main result is that for any asymptotic cone of mcg S the maximal dimension of locally compact subsets coincides with the maximal rank of free abelian subgroups of MCG S . An application is a proof of Brock-Farb s Rank Conjecture which asserts that MCG S has quasi-flats of dimension N if and only if it has a rank N free abelian subgroup. Hamenstadt has also given a proof of this conjecture using different methods. We also compute the maximum dimension of quasi-flats in Teichmuller space with the Weil-Petersson metric. Introduction The coarse geometric structure of a finitely generated group can be studied by passage to its asymptotic cone which is a space obtained by a limiting process from sequences of rescalings of the group. This has played an important role in the quasi-isometric rigidity results of DS KaL KlL and others. In this paper we study the asymptotic cone Mm S of the mapping class group of a surface of finite type. Our main result is Dimension Theorem. The maximal topological dimension of a locally compact subset of the asymptotic cone of a mapping class group is equal to the maximal rank of an abelian subgroup. Note that BLM showed that the maximal rank of an abelian subgroup of a mapping class group of a surface with negative Euler characteristic is 3g 3 p where g is the genus and p the number of boundary components. This is also the number of components of a pants decomposition and hence the largest rank of a pure Dehn twist subgroup. First author supported by NSF grants DMS-0091675 and DMS-0604524. Second author supported by NSF grant DMS-0504019. 1056 JASON A. BEHRSTOCK AND YAIR N. MINSKY As an

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