Back to WTC Summer 2006
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
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.
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
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.
Stable homology of automorphisms of free
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.
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
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
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
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.
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).
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.)