Lee Mosher

Mapping class groups are automatic

Let S be a compact surface, possibly with the extra structure of an orientation or a finite set of distinguished points called punctures. The mapping class group of S is the group MCG(S) = Homeo(S)/ Homeo0(S), where Homeo(S) is the group of all homeomorphisms of S preserving the extra structure, and Homeo0(S) is the normal subgroup of all homeomorphisms isotopic to the identity through elements of Homeo(S). By convention each boundary component of S contains a puncture; in general, if a boundary component contains no puncture it may be collapsed to a puncture without changing the mapping class group. Given a group G, suppose that A = {g1, . . . , gk} is a finite set of generators and L is a set of words over the alphabet A, such that each element of G is represented by at least one word in L, and L is a regular language over A, i.e. one can check for membership in L with a finite automaton. The words in L representing a given group element can be thought of as normal forms for that element. Then L is an automatic structure for G if for any two words v, w ∈ L, one can check with a finite automaton whether the associated group elements v̄ and w̄ are equal, and whether they differ by a certain generator. A more geometric characterization of automaticity is given by the fellow traveller property , which says that there is a constant K such that for any v, w ∈ L, if d(v̄, w̄) ≤ 1, where d(v̄, w̄) is the word length of v̄−1w̄, then for any n ≥ 0, letting v(n) and w(n) be the prefixes of v̄ and w̄ of length n, then d(v̄(n), w̄(n)) ≤ K. A group G is automatic if it has an automatic structure. The theory of automatic groups is presented in [ECHLPT]. An automatic group has a quadratic isoperimetric inequality, and a quadratic time algorithm for the word problem, in addition to many other nice geometric and computational properties.

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.

The free splitting complex of a free group, I Hyperbolicity

We prove that the free splitting complex of a finite rank free group, also known as Hatchers sphere complex, is hyperbolic.

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.

A hyperbolic-by-hyperbolic hyperbolic group

Given a short exact sequence of finitely generated groups

The geometry of surface-by-free groups

Abstract. We show that every word hyperbolic, surface-by-(noncyclic) free group

The expansion factors of an outer automorphism and its inverse

\Gamma

On train-track splitting sequences

is as rigid as possible: the quasi-isometry group of

Lipschitz retraction and distortion for subgroups of Out(Fn)

\Gamma