Dimension and rank for mapping class groups

By Jason A. Behrstock, Yair N. Minsky. 2005

93 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/341cb9af1469212635cb3354006e6bbb95af23bf

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Quasiflats in hierarchically hyperbolic spaces (2017, background): “embedded in a given mapping class group if n is at most the complexity of the surface [BM08a], this established the dimension of the top-dimensional quasiflats in the mapping class group.”
Coarse and synthetic Weil–Petersson geometry: quasi-flats, geodesics and relative hyperbolicity (2007, methodology): “ a conjecture of the ﬁrst author and Farb [BF] in the case ζ(S)=3. We remark that the following statement has been obtained independently (and for all surfaces S) by work of Behrstock and Minsky (see [BM]). Corollary 4. (GEOMETRIC RANK) When ζ(S)=3 the geometric rank of the pants graph P(S)and hence”
Geometry and rigidity of mapping class groups (2008, methodology): “The main references are Masur and Minsky [40] and Behrstock and Minsky [8].”
The Weil-Petersson metric geometry (2007, unknown): “In collaborations Behrstock-Minsky [BM06] and Behrstock-Kleiner-MinskyMosher have been investigating the asymptotic cones AC of MCG (the Gromov-Hausdorff limits of rescalings of the group word-metric.)”
Dimension of asymptotic cones of Lie groups (2007, unknown): “c2]. Here we focus on the covering dimension of the asymptotic cone, which was considered in Gromov’s book [Gro, Chap. 2], in Burillo’s paper [Bur] and in [LePi] in the context of solvable groups, in [BeMi] in the case of the mapping class group, in [DrSm] in connection with asymptotic Assouad-Nagat”
Geometry of the mapping class groups III: Quasi-isometric rigidity (2005, background): “gical dimension of an asymptotic cone for M(S). Farb, Lubotzky and Minsky [FLM01] showed that the geometric rank of M(S) is at least 3g− 3 + m; thus as an immediate corollary of our theorem we obtain [BM05]. Corollary[BM05]: The geometric rank of M(S) equals 3g−3+m. The organization of this paper is”
Asymptotic Geometry of the Mapping Class Group and Teichmuller Space (2005, background): “This readily implies that these directions are quasi-geodesically stable, ie, for the quasi-geodesic axis given by taking powers of a pseudo-Anosov, there is a bound M.”