Mladen Bestvina

Bounded cohomology of subgroups of mapping class groups

We show that every subgroup of the mapping class group MCG(S )o f ac ompact surface S is either virtually abelian or it has innite dimensional second bounded cohomology. As an application, we give another proof of the Farb{ Kaimanovich{Masur rigidity theorem that states that MCG(S )d oes not contain a higher rank lattice as a subgroup.

Laminations, trees, and irreducible automorphisms of free groups

Abstract. We examine the action of Out(Fn) on the set of (irreducible) laminations. Consequences include a special case of the Tits alternative for Out(Fn), the discreteness of certain naturally arising group actions on trees, and word hyperbolicity of certain semidirect products.

The Tits alternative for Out (F~n) I: Dynamics of exponentially-growing automorphisms

The Tits alternative for Out(F_n) is reduced to the case where all elements in the subgroup under consideration grow polynomially.

ℝ-Trees in Topology, Geometry, and Group Theory

This paper is intended as a survey of the theory and applications of real trees from a topol-ogists point of view. The idea of an all-inclusive historical account was quickly abandoned at the start of this undertaking, but I hope to describe the main ideasin the subject with emphasis on applications outside the theory of ℝ-trees. The “Rips machine”, i.e., the classification of measured laminations on 2-complexes, is the key ingredient. Roughly speaking, the Rips machine is an algorithm that takes as input a finite 2-complex equipped with a transversely measured lamination (more precisely, a band complex), and puts it in a “normal form”.

Non-positively curved aspects of Artin groups of finite type

Artin groups of nite type are not as well understood as braid groups. This is due to the additional geometric properties of braid groups coming from their close connection to mapping class groups. For each Artin group of nite type,

The asymptotic geometry of right-angled Artin groups, I

We show that if X is a piecewise Euclidean 2-complex with a cocompact isometry group, then every 2-quasiflat in X is at finite Hausdorff distance from a subset Q which is locally flat outside a compact set, and asymptotically conical.

Solvable subgroups of Out(Fn) are virtually Abelian

Let Fn be the free group of rank n, let Aut(Fn) be its automorphism group and let Out(Fn) be its outer automorphism group. We show that every solvable subgroup of Out(Fn) has a finite index subgroup that is finitely generated and free Abelian. We also show that every Abelian subgroup of Out(Fn) has a finite index subgroup that lifts to Aut(Fn).

The boundary of the complex of free factors

We give a description of the boundary of a complex of free factors that is analogous to E. Klarreichs description of the boundary of a curve complex. The argument uses the geometry of folding paths developed by Bestvina and Feighn as well as structural results about very small trees developed by Coulbois, Hilion, Lustig, and Reynolds.

Limit Groups are Cat(0)

We prove that every limit group acts geometrically on a CAT(0) space with the isolated flats property.

Dimension of the Torelli group for Out(Fn)

Let