Back to WTC Summer 2006

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Peter Brinkmann
Algorithmically improving train tracks
I will report on recent work motivated by the solution to the conjugacy problem in free-by-cyclic groups due to Bogopolski, Martino, Maslakova, and Ventura. Specifically, the goal is to construct a version of improved relative train tracks (akin to those of Bestvina, Feighn, and Handel) in an algorithmic fashion, and to use these properties to obtain generalizations of algorithmic results.

Tullia Dymarz
Tukia's theorem and boundary theory for solvable groups
We prove a foliated version of Tukia's theorem on uniformly quasiconformal groups for boundaries of certain solvable groups.

Eskin-Fisher-Whyte recently proved quasi-isometric rigidity for a wide class of polycyclic groups. One of the ingredients in their proof is our version of Tukia's theorem.

The talk will focus mostly on describing the geometry of these solvable groups and explaining the statement and proof idea of the theorem.

Anna Lenzhen
Teichmuller geodesics that do not have a limit in PMF
We construct a Teichmuller geodesics which does not have a limit on the Thurston boundary of Teichmuller space.

Lars Louder
Accessibility of Limit Groups
To a limit group one can associate hierarchies of splittings over abelian and cyclic subgroups. Sela proved that the hierarchy of splittings over cyclic subgroups, the so called "cyclic analysis lattice," is finite for any limit group. We give an easier proof of finiteness of both the abelian and cyclic analysis lattices, with an eye toward Sela's conjecture that there is a notion of Krull dimension for varieties defined over the free group.

Soren Galatius
Stable homology of automorphisms of free groups
The homology H_k(Aut(F_n)) of the automorphism group of a free group is known to be independent of n, as long as n > 2k+1. I will explain how to determine the homology in this stable range. The answer is that the homology agrees with the homology of the space QS^0, i.e. the direct limit of the n-fold loop space of the n-sphere, as n goes to infinity. The proof uses graphs and outer space, and is homotopy theoretic in flavor.

Howard Masur
Ergodic theory of translation surfaces
Let X be a closed Riemann surface and omega a holomorphic 1 form on X. The pair (X, \omega) defines the structure of a translation surface. This structure is equivalent to one htat is given by a collection of polygons in the plane that are glued along their boundaries by translations. for each direction theta, there is a flow in direction theta by straight lines on the surface. In genus one this gives the well known linear flow on the torus. In higher genus there are many additional interesting phenomena. This talk will survey what is known about the topological properties and ergodic theory of these flows.

Alexandra Pettet
Cohomology of some subgroups of the automorphism groups of the free group
IA_n is the subgroup of Aut(F_n) which acts trivially on the homology of the free group and is thus an analogue for the Torelli subgroup of the mapping class group of a surface. We implement different methods for studying cohomology of IA_n and one of its subgroups.

Ben Schmidt
Blocking light in compact Reimannian manifolds
(Joint with J. Lafont) To what extent does the collision of light determine the geometry of space? With this question in mind, I'll discuss two conjectures (and supporting results) asserting that compact Riemannian manifolds with light behaving similarly to light in a compact locally symmetric space are necessarily isometric to a compact locally symmetric space.
  
Juan Souto
Heegard splittings and minimal surfaces
We discuss the proof of a theorem due to Pitts and Rubinstein that ensure that under some reasonable conditions surfaces in 3-manifolds which decompose the manifold into two handlebodies are isotopic to minimal surfaces.

Kevin Wortman
A finitely-presented solvable group with a small quasi-isometry group
I'll present an example of an infinite, finitely-presented solvable group whose quasi-isometry group is a Lie group (over local fields).

Unipotent flows on moduli space for genus 2
I'll talk about some advances in the classification of probability measures that are ergodic with respect to unipotent flows on the moduli space of abelian differentials in genus 2. (Joint with Kariane Calta.)