Algom-Kfir
- Mapping tori of small dilatation
automorphisms
A self map of a graph is an irreducible train-track map if
positive iterates of edges are immersed edge paths, and every edge has
an iterate that covers any other edge. To an irreducible train-track
(ITT) map one can attach a real number \lam>1 called the dilatation
of f. Among all \phi \in Out(F_n) that can be represented by an ITT
map, the smallest possible dilatation such maps is on the order of
2^{1/n}. We say that f is P-small if its dilatation is on the order of
P^{1/n}. We show that there is a finite number of mapping tori X_1, ...
, X_k of self maps of graphs, so that the mapping torus of a P-small
itt map can be obtained by surgery from one of the X_i's. As a
corollary we prove that the fundamental group of such a mapping torus
has a presentation with boundedly many generators and relations and the
bound depends only on P. This is joint work with Kasra Rafi.
Avramidi
- Symmetries of aspherical manifolds
In this talk, I will describe methods for bounding isometry
groups of Riemannian metrics on some aspherical manifolds and of the
lifted metrics on their universal covers. The theme is that topological
properties of an aspherical manifold can restrict the symmetries of an
arbitrary complete Riemannian metric on that manifold. I will
illustrate this by explaining why on a finite volume irreducible
locally symmetric manifold, no metric has more symmetry than the
locally symmetric metric.
Chaika
- Quantitative shrinking targets for
IETs and rotations
In this talk we present some quantitative shrinking target
results. Consider $T:[0,1] \to [0,1]$. One can ask how quickly under T
a typical point $x$ approaches a typical point $y$. In particular given
$\{a_i\}_{i=1}^{\infty}$ is $T^ix \in B(y,a_i)$ infinitely often? A
finer question of whether $T^ix \in B(y,a_i)$ as often as one would
expect will be discussed. That is, does
$$\underset{N \to \infty}{\lim}\frac{\underset{n=1}{\overset{N}{\sum}}
\chi_{B(y,a_n)(T^nx)}}{\underset{n=1}{\overset{N}{\sum}} 2a_n}=1$$
for almost every $x$.
These results also apply to a billiard in the typical direction of any
rational polygons. This is joint work with David Constantine.
Gadre
- Word length statistics for
Teichmüller geodesics and singularity of harmonic measures
Kaimanovich and Masur showed that a random walk on the
mapping class group when projected to Teichmüller space by its action
converges almost surely to the Thurston boundary. This defines a
harmonic or hitting measure on the boundary. Restricting to finitely
supported random walks, we consider word length statistics along
Teichmüller geodesics. We show that along geodesics typical with
respect to harmonic measure word length grows linearly. On the other
hand, we also show that along geodesics typical with respect to
Lebesgue measure word length grows super-linearly. In particular, this
gives a geometric proof of the fact that harmonic measure is singular
with respect to Lebesgue measure. This is joint work with J. Maher and
G. Tiozzo. We also discuss other contexts in which such results hold.
Hironaka
- Quasi-periodic mapping classes
A pseudo-Anosov mapping class is said to have P-small
dilatation if its normalized dilatation (dilatation raised to the
absolute value of the topological Euler characterisitc) is bounded by
P. We examine the question of whether P-small dilatations must be
"nearly" periodic. More precisely, we define the notions of
weakly and strongly quasi-periodic mapping classes, we give two large
classes of examples of this type, and show that these examples arise
naturally as deformations of mapping classes on fibered faces.
Kapovich
- On the Cannon-Thurston map for
hyperbolic free-by-cyclic groups
Let
$\Phi\in Aut(F_N)$ be an atoroidal iwip automorphism of a free group
$F_N$, $N\ge 3$ and let $M_\Phi=F_N\rtimes_\Phi <t>$ be the
mapping torus group of $\Phi$. The group $M_\Phi$ is Gromov-hyperbolic
and it follows from the work of Mitra that the inclusion $\iota:F_N\to
M_\Phi$ extends to a continuous surjective map $\hat \iota: \partial
F_N\to\partial M_\Phi$, called the \emph{Cannon-Thurston map}. We study
the fibers of the map $\hat \iota$.
We prove that for any $\Phi$ as above, the map $\hat \iota$ is
finite-to-one and that the preimage of every point of $\partial M_\Phi$
has cardinality $\le 2N$.
We also prove that every point $S\in \partial M_\Phi$ with $\ge 3$
preimages in $\partial F_N$ is rational and has the form
$(xt^m)^\infty$ where $x\in F_N, m\ne 0$, and that there are at most
$4N-5$ $F_N$-orbits of such points in $\partial M_\Phi$ (for the
translation action of $F_N$ on $\partial M_\Phi$).
We show that, by contrast, for $k=1,2$ there are uncountably many
points $S\in \partial M_\Phi$ with exactly $k$ preimages in $\partial
F_N$. The talk is based on a joint paper with Martin Lustig.
M.
Lee - Dynamics on the
PSL(2,C)-character variety of certain hyperbolic 3-manifolds
The PSL(2,C)-character variety of a hyperbolic 3-manifold M
is essentially the set of homomorphisms of the fundamental group of M
into PSL(2,C), up to conjugacy. We will discuss the action of the
group of outer automorphisms of the fundamental group of M on this
space. In particular, we will discuss how one can find domains of
discontinuity for the action.
S.
R. Lee - Variation of Houghton's
groups and its applications
Houghton's groups {H_n} is a family of amenable groups with
the property that H_n has type FP_n-1 but not FP_n. They have potential
to provide solutions for the following questions.
Q1: If a finitely presented group has in finite virtual first Betti
number then must it be large? Does it contain F_2?
Q2: Are there groups of type F_n-1(FP_n-1) but not F_n (FP_n) which do
not contain Z^2 subgroup (n >3) ?
We study `Extended' Houghton's groups and `Twisted' Houghton's groups
to answer those questions. As main tools, we use CAT(0) cubical
complexes on which Houghton's groups act as well as a combination of
Bestvina-Brady Morse theory and Ken Brown's criterion for finiteness
properties of groups. The first part is joint work with Nir Lazarovich
and the second part is joint work with Noel Brady.
McReynolds
- Distinguishing metrics via
spectral properties
One hopes in geometry that special metrics should be
distinguished among all other metrics via some geometric property or
invariant. These types of rigidity results are extremely difficult to
prove. The next best thing to prove is this rigidity holds for near by
metrics. On the other hand, we have fundamental results like Weyl's Law
and Margulis' thesis on the coarse growth rate of the number of
eigenvalues for the Laplacian or the number of primitive closed
geodesics that hold for reasonably broad classes of metrics. In this
talk, I will introduce a finer structural condition on the set of
primitive geodesic lengths that when present gives additional
qualitative properties. The primary goal for this condition is to
single out metrics arising from arithmetic constructions on manifolds
that support locally symmetric metrics. This is joint work with
Jean-Francois Lafont.
Moore
- Nonassociative Ramsey theory and
the amenability problem for Thompson's group
In 1973, Richard Thompson considered the question of whether his
newly defined group F was
amenable. The motivation for this problem stemed from his observation
--- later rediscovered by Brin and Squire --- that F did not contain a free group on
two generators, thus making it a candidate for a counterexample to the
von Neumann-Day problem. While the von Neumann-Day problem was solved
by Ol'shanskii in the class of finitely generated groups and
Ol'shanskii and Sapir in the class of finitely presented groups, the
question of F's amenability
was sufficiently basic so as to become of interest in its own right.
In this talk, I will analyze this problem from a Ramsey-theoretic
perspective. In particular, the problem is related to generalizations
of Ellis's Lemma and Hindman's Theorem to the setting of nonassociative
binary systems. The amenability of F
is itself equivalent to the existence of certain finite Ramsey numbers.
I will also discuss the growth rate of the F\olner function for F (if it exists).
Rafi
- Geometry of Teichmüller space
We review recent results about the Teichmüller space
equipped with the Teichmüller metric. We give an inductive description
of a Teichmüller geodesic using the Teichmüller geodesics of surfaces
with lower complexity. We use these results to compare how the
Teichmüller space is similar or different from the hyperbolic space.