Dimension and Rank for Mapping Class Groups

We study the large scale geometry of the mapping class group, MCG. Our main result is that for any asymptotic cone of MCG, the maximal dimension of locally compact subsets coincides with the maximal rank of free abelian subgroups of MCG. An application is a proof of Brock-Farb’s Rank Conjecture which asserts that MCG 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 quasiflats in Teichmuller space with the Weil-Petersson metric. 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], [KL1] [KL2], and others. In this paper we study the asymptotic cone M(S) of the mapping class group of a surface of finite type. Our main result is Dimension Theorem. The maximal topological dimension of a locallycompact 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. As an application we obtain a proof of the “geometric rank conjecture” for mapping class groups, formulated by Brock and Farb [BF], which states: Rank Theorem. The geometric rank of the mapping class group of a surface of finite type is equal to the maximal rank of an abelian subgroup. Hamenstädt has previously announced a proof of the rank conjecture for mapping class groups, which has now appeared in [Ham]. Her proof uses the geometry of train tracks and establishes a homological version of the First author partially supported by NSF grant 0091675. Second author supported by NSF grant DMS-0504019. 1 2 JASON A. BEHRSTOCK AND YAIR N. MINSKY dimension theorem. Our methods are quite different from hers, and we hope that they will be of independent interest. The geometric rank of a group G is defined as the largest n for which there exists a quasi-isometric embedding Z → G, also known as an n-dimensional quasi-flat. It was proven in [FLM] that, in the mapping class group, maximal rank abelian subgroups are quasi-isometrically embedded—thereby giving a lower bound on the geometric rank. This was known when the Rank Conjecture was formulated, thus the conjecture was that the known lower bound for the geometric rank is sharp. The affirmation of this conjecture follows immediately from the dimension theorem and the observation that a quasi-flat, after passage to the asymptotic cone, becomes a bi-Lipschitzembedded copy of R. We note that in general the maximum rank of (torsion-free) abelian subgroups of a given group does not yield either an upper or a lower bound on the geometric rank of that group. For instance, non-solvable BaumslagSolitar groups have geometric rank one [Bur], but contain rank two abelian subgroups. To obtain groups with geometric rank one, but no subgroup isomorphic to Z, one may take any finitely generated infinite torsion group. The n-fold product of such a group with itself has n-dimensional quasi-flats, but no copies of Z. Similar in spirit to the above results, we also prove: Dimension Theorem for Teichmüller space. Every locally-compact subset of an asymptotic cone of Teichmüller space with the Weil-Petersson metric has topological dimension at most ⌊ 2 ⌋. The dimension theorem implies the following, which settles another conjecture of Brock–Farb. Rank Theorem for Teichmüller space. The geometric rank of the WeilPetersson metric on the Teichmüller space of a surface of finite type is equal to ⌊ 2 ⌋. This conjecture was made by Brock–Farb after proving this result in the case ⌊ 2 ⌋ ≤ 1, by showing that in such cases Teichmüller space is δhyperbolic [BF]. (Alternate proofs of this result were obtained in [Be2] and [Ara].) We also note that the lower bound on the geometric rank of Teichmüller space is obtained in [BF]. Outline of the proof For basic notation and background see §1. We will define a family P of subsets of M(S) with the following properties: Each P ∈ P comes equipped with a bi-Lipschitz homeomorphism to a product F ×A, where (1) F is an R-tree (2) A is the asymptotic cone of the mapping class group of a (possibly disconnected) proper subsurface of S. DIMENSION AND RANK FOR MAPPING CLASS GROUPS 3 There will also be a Lipschitz map πP : M (S) → F such that: (1) The restriction of πP to P is projection to the first factor. (2) πP is locally constant in the complement of P . These properties immediately imply that the subsets {t}×A in P = F ×A separate M(S) globally. The family P will also have the property that it separates points, that is: for every x 6= y in M(S) there exists P ∈ P such that πP (x) 6= πP (y). Using induction, we will be able to show that locally compact subsets of A have dimension at most r(S) − 1, where r(S) is the expected rank for M(S). The separation properties above together with a short lemma in dimension theory then imply that locally compact subsets of M(S) have dimension at most r(S). Section 1 will detail some background material on asymptotic cones and on the constructions used in Masur-Minsky [MM1, MM2] to study the coarse structure of the mapping class group. Section 2 introduces product regions in the group and in its asymptotic cone which correspond to cosets of curve stabilizers. Section 3 introduces the R-trees F , which were initially studied by Behrstock in [Be2]. The regions P ∈ P will be constructed as subsets of the product regions of Section 2, in which one factor is restricted to a subset which is one of the R-trees. The main technical result of the paper is Theorem 3.5, which constructs the projection maps πP and establishes their locally-constant properties. An almost immediate consequence is Theorem 3.6, which gives the family of separating sets whose dimension will be inductively controlled. Section 4 applies Theorem 3.6 to prove the Dimension Theorem. Section 5 applies the same techniques to prove a similar dimension bound for the asymptotic cone of the Teichmüller space with its Weil-Petersson metric, and to deduce a corresponding geometric rank statement. Acknowledgements. The authors are grateful to Lee Mosher for many insightful discussions, and for a simplification to the original proof of Theorem 3.5. We would also like to thank Benson Farb for helpful comments on an earlier draft.