# Abstracts

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.