Benson Farb

A primer on mapping class groups

Given a compact connected orientable surface S there are two fundamental objects attached: a group and a space. The group is the mapping class group of S, denoted by Mod(S). This group is defined by the isotopy classes of orientation-preserving homeomorphism from S to itself. Equivalently, Mod(S) may be defined using diffeomorphisms instead of homeomorphisms or homotopy classes instead of isotopy classes. The space is the Teichmüller space of S, Teich(S). Teichmüller space and moduli space are fundamental objects in fields like low-dimensional topology, algebraic geometry and mathematical physics. If X (S) < 0, the Teichmüller space can be thought of as the set of homotopy classes of hyperbolic structures of S or, equivalently, as the set of isotopy classes of hyperbolic metrics on S, HypMet(S). The group and the space are connected through the moduli space in the following way. The group of orientation-preserving diffeomorphisms of S, Diff+(S) acts on HypMet(S) and this action descends to an action of Mod(S) on Teich(S) which is properly discontinuous. The quotient space,

FI-modules and stability for representations of symmetric groups

In this paper we introduce and develop the theory of FI-modules. We apply this theory to obtain new theorems about: - the cohomology of the configuration space of n distinct ordered points on an arbitrary (connected, oriented) manifold - the diagonal coinvariant algebra on r sets of n variables - the cohomology and tautological ring of the moduli space of n-pointed curves - the space of polynomials on rank varieties of n x n matrices - the subalgebra of the cohomology of the genus n Torelli group generated by H^1 and more. The symmetric group S_n acts on each of these vector spaces. In most cases almost nothing is known about the characters of these representations, or even their dimensions. We prove that in each fixed degree the character is given, for n large enough, by a polynomial in the cycle-counting functions that is independent of n. In particular, the dimension is eventually a polynomial in n. In this framework, representation stability (in the sense of Church-Farb) for a sequence of S_n-representations is converted to a finite generation property for a single FI-module.

FI-modules over Noetherian rings

FI-modules were introduced by the first three authors to encode sequences of representations of symmetric groups. Over a field of characteristic 0, finite generation of an FI-module implies representation stability for the corresponding sequence of Sn ‐representations. In this paper we prove the Noetherian property for FI-modules over arbitrary Noetherian rings: any sub-FI-module of a finitely generated FI-module is finitely generated. This lets us extend many results to representations in positive characteristic, and even to integral coefficients. We focus on three major applications of the main theorem: on the integral and mod p cohomology of configuration spaces; on diagonal coinvariant algebras in positive characteristic; and on an integral version of Putman’s central stability for homology of congruence subgroups. 20B30; 20C32

Rank-1 phenomena for mapping class groups

Let Σg be a closed, orientable, connected surface of genus g ≥ 1. The mapping class group Mod(Σg) is the group Homeo(Σg)/Homeo0(Σg) of isotopy classes of orientation-preserving homeomorphisms of Σg. It has been a recurring theme to compare the group Mod(Σg) and its action on the Teichmuller space T (Σg) to lattices in simple Lie groups and their actions on the associated symmetric spaces. Indeed, the groups Mod(Σg) share many of the properties of (arithmetic) lattices in semisimple Lie groups. For example they satisfy the Tits alternative, they have finite virtual cohomological dimension, they are residually finite, and each of their solvable subgroups is polycyclic. A well-known dichotomy among the lattices in simple Lie groups is between lattices in rank one groups and higher-rank lattices, i.e. those lattices in simple Lie groups of R-rank at least two. It is somewhat mysterious whether Mod(Σg) is similar to the former or the latter. Some higher rank behavior of Mod(Σg) is indicated by the cusp structure of moduli space, by the fact that Mod(Σg) has Serre’s property (FA) [CV], and by Ivanov’s version (see, e.g. [Iv2]) for Mod(Σg) of Tits’s Theorem on automorphism groups of higher rank buildings. In this note we add two more properties to the list (see, e.g. [Iv1, Iv2, Iv3] and the references therein) of properties which exhibits similarities of Mod(Σg) with lattices in rank one groups: every infinite order element of Mod(Σg) has linear growth in the word metric, and Mod(Σg) is not bound∗Supported in part by NSF grant DMS 9704640 and by a Sloan Foundation fellowship. †Supported in part by the US-Israel BSF grant. ‡Supported in part by NSF grant DMS 9971596

Convex cocompact subgroups of mapping class groups

We develop a theory of convex cocompact subgroups of the mapping class group MCG of a closed, oriented surface S of genus at least 2, in terms of the action on Teichmuller space. Given a subgroup G of MCG defining an extension 1 → π 1 (S) → T G → G → 1, we prove that if Γ G is a word hyperbolic group then G is a convex cocompact subgroup of MCG. When G is free and convex cocompact, called a Schottky subgroup of MCG, the converse is true as well; a semidirect product of π 1 (S) by a free group G is therefore word hyperbolic if and only if G is a Schottky subgroup of MCG. The special case when G = Z follows from Thurstons hyperbolization theorem. Schottky subgroups exist in abundance: sufficiently high powers of any independent set of pseudo-Anosov mapping classes freely generate a Schottky subgroup.

On the asymptotic geometry of abelian-by-cyclic groups

Gromov’s Polynomial Growth Theorem [Gro81] states that the property of having polynomial growth characterizes virtually nilpotent groups among all finitely generated groups. Gromov’s theorem inspired the more general problem (see, e.g. [GdlH91]) of understanding to what extent the asymptotic geometry of a finitelygenerated solvable group determines its algebraic structure—in short, are solvable groups quasi-isometrically rigid? In general they aren’t: very recently A. Dioubina [Dio99] has found a solvable group which is quasi-isometric to a group which is not virtually solvable; these groups are finitely generated but not finitely presentable. In the opposite direction, first steps in identifying quasi-isometrically rigid solvable groups which are not virtually nilpotent were taken for a special class of examples, the solvable BaumslagSolitar groups, in [FM98] and [FM99b]. The goal of the present paper is to show that a much broader class of solvable groups, the class of finitely-presented, nonpolycyclic, abelian-bycyclic groups, is characterized among all finitely-generated groups by its quasi-isometry type. We also give a complete quasi-isometry classification of the groups in this class; such a classification for nilpotent groups remains a major open question. Motivated by these results, we offer a conjectural picture of quasi-isometric classification and rigidity for polycyclic abelianby-cyclic groups in §10.1.

Quasi-flats and rigidity in higher rank symmetric spaces

In this paper we use elementary geometrical and topological methods to study some questions about the coarse geometry of symmetric spaces. Our results are powerful enough to apply to noncocompact lattices in higher rank symmetric spaces, such as SL(n, 2), n > 3: Theorem 8.1 is a major step towards the proof of quasiisometric rigidity of such lattices ([E]). We also give a different, and effective, proof of the theorem of Kleiner-Leeb on the quasi-isometric rigidity of higher rank symmetric spaces ([KL]).

Quasi-isometric rigidity for the solvable Baumslag-Solitar groups, II

Let BS(1,n)= . We prove that any finitely-generated group quasi-isometric to BS(1,n) is (up to finite groups) isomorphic to BS(1,n). We also show that any uniform group of quasisimilarities of the real line is bilipschitz conjugate to an affine group.

Groups of homeomorphisms of one-manifolds III: nilpotent subgroups

Plante-Thurston proved that every nilpotent subgroup of

Problems on Mapping Class Groups and Related Topics

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