Abstracts of Talks
We discuss the relation between coarse geometric properties of the spine of outer space and metric properties of the space of triples of pairwise distinct measured laminations. It turns out that the geometry of triples is closely related to the symmetrized shortest-Lipschitz-constant metric. As applications, we find paths in the splitting graph and construct quasi-morphisms for subgroups of Out(Fn) which are not virtually abelian.
We study the shape of the moduli space of a surface of finite type. In particular, we compute the asymptotic behavior of the Teich diameter of the thick part of the moduli space. For a surface S of genus g with b boundary components define the complexity of S to be 3g-3+b. We show that the diameter grows like logarithm of the complexity.
I will recall the definition of the Lipschitz metric on Outer space and its basic properties. I will then outline an alternative proof of the existence of train track representatives for irreducible automorphisms. The proof is analogous to the Bers' proof of the Nielsen-Thurston classification of surface homeomorphisms.
By a theorem of Thurston, in the subgroup of the mapping class group generated by Dehn twists around two curves which fill, every element not conjugate to a power of one of the twists is pseudo-Anosov. We prove an analogue of this theorem for the outer automorphism group of a free group. This is joint work with Alexandra Pettet.
By definition, a geodesic current on a free group FN is a locally finite F N-invariant Borel measure on ∂ FN × ∂ FN — diag. The notion of a geodesic current is a measure-theoretic analog of the notion of the conjugacy class of a nontrivial element of a free group (or the conjugacy class of an infinite cyclic subgroup of FN) and the space of currents on FN is a natural companion of Culler-Vogtmann's Outer space. We introduce and study the space of "generalized currents" on FN which are measure-theoretic analogs of conjugacy classes of finitely generated subgroups of FN Generalized currents turn out to provide a natural model for working with "generalized words", i.e. words that are not written on a linear domain; finding such models is an important task in Combinatorics.
Yael Algom Kfir
The Lipschitz metric on Outer Space is not symmetric. In fact d(x,y)/d(y,x) can be arbitrarily large. In joint work with Mladen Bestvina, we define a piecewise differentiable function ψ on Outer Space (which is constant on the orbits of Out(Fn) and show that d(x,y) can be bounded in terms of d(y,x) and |ψ(x) - ψ(y)|. I will discuss the proof of this theorem and some applications.
For a closed surface, the hyperelliptic Torelli group is the group of isotopy classes of self-homeomorphisms that act trivially on homology, and that commute with some fixed hyperelliptic involution. Hain has conjectured that this group is generated by Dehn twists about separating curves that are fixed by the involution. In joint work with Tara Brendle, we prove the inductive step needed for the conjecture, a Birman exact sequence for the hyperelliptic Torelli group. We also give factorizations of many simple elements of the hyperelliptic Torelli group into Hain's generators.
Right-angled Artin groups interpolate between free groups and free abelian groups. Thus, their (outer) automorphism groups can be viewed as interpolating between Out(Fn) and GL(n,Z) To study these groups one would like to construct an analogue of outer space for any right-angled Artin group. We discuss various approaches, some progress, and some hurdles to constructing such a space. (Joint work with Karen Vogtmann)
J.H.C. Whitehead thought about automorphisms of free groups by thinking of them as diffeomorphisms of a doubled handlebody. Outer space can be identified with weighted sphere systems in such a doubled handlebody. This model has been used to prove various theorems about Out(Fn), such as bounding its isoperimetric functions, and to study various Out(Fn)-complexes, such as the complexes of free factorizations of Fn or the complex of free factors of Fn. I will give a tour of this model and some of its features.
Joint work with Mladen Bestvina. For any finite collection fi of fully irreducible automorphisms of the free group Fn we construct a connected δ-hyperbolic Out(Fn)-complex in which each fi has positive translation length.
We prove that there exist normal free subgroups of Out(Fn) and of the mapping class group
all of whose non-trivial elements are irreducible(iwip)/pseudo-anosov.
This is done using small cancellation techniques on
Bestvina-Feighn's hyperbolic complex for Out(Fn) and on the complex of curves.
This is a joint work with Francois Dahmani.
We show for k at least 3, that for any matrix A in GL(k,Z), there is a hyperbolic fully irreducible automorphism of the free group of rank k whose induced action on the rank k free abelian group is given by the matrix A. This is joint work with Matt Clay.
(Joint work with Lee Mosher) I will talk about our proof that for any subgroup H of Out(Fn), either H has a finite index subgroup that fixes the conjugacy class of some proper, nontrivial free factor of Fn or H contains a fully irreducible element.
shall sketch the proof of several results that constrain the way in which mapping class groups and automorphism groups of free groups can act by isometries on CAT(0) spaces. I shall discuss consequences concerning the linear representations of these groups as well as maps between them.
I will prove that Out(Fn) is the full group of automorphisms of the free splitting graph.