**Title**: Divergence, thick groups, and morse geodesics

**Abstract**: In a metric space the divergence of a pair of rays is a way

to measure how quickly they separate from each other. Understanding

what divergence rates are possible in the presence of non-positive

curvature was raised as a question by Gromov and then refined by

Gersten. We will describe a construction of groups with several

interesting properties, some of which shed light on the above

question. (Joint work with Cornelia Drutu.)

Abstracts:

John Pardon (3:30):

Totally disconnected groups (not) acting on three-manifolds.

Hilbert´s Fifth Problem asks whether every topological group which is a manifold is in fact a (smooth!) Lie group; this was solved inthe affirmative by Gleason and Montgomery–Zippin. A stronger conjecture is that a locally compact topological group which acts faithfully on a manifold must be a Lie group. This is the Hilbert–Smith Conjecture,which in full generality is still wide open. It is known, however (as a corollary to the work of Gleason and Montgomery–Zippin) that it suffices to rule out the case of the additive group of p-adic integers acting faithfully on a manifold. I will present a solution in dimension three. The proof uses tools from low-dimensional topology, for example incompressible surfaces, minimal surfaces, and a property of the mapping class group.

Larry Guth (5:00):

Contraction of areas and homotopy type of mappings.

I´m going to talk about connections between the geometry of a map and its homotopy type. Suppose that we have a map from the unit m-sphere to the unit n-sphere. We say that the k-dilation of the map is < L if each k-dimensional surface with k-dim volume V is mapped to an image with k-dim volume at most LV. Informally, if the k-dilation of a map is less than a small epsilon, it means the map strongly shrinks each k-dimensional surface. Our main question is: can a map with very small k-dilation still be homotopically non-trivial?

Here are the main results. If k > (m+1)/2, then there are homotopically non-trivial maps from S^m to S^{m-1} with arbitrarily small k-dilation. But if k is at most (m+1)/2, then every homotopically non-trivial map from S^m to S^{m-1} has k-dilation at least c(m) > 0.

Title: Statistics for Teichmuller geodesics.

Abstract: We describe two ways of picking a geodesic “at random” in a

space, one coming from the standard Lebesgue measure on the visual sphere, and the other coming from random walks. The spaces we’re interested in are hyperbolic space and Teichmuller space, together with some discrete group action on the space. We investigate the growth rate of word length as you move along the geodesic, and we show these growth rates are different depending on how you choose the geodesic. This is joint work with Vaibhav Gadre and Giulio Tiozzi.

Abstract: Goldman and Turaev discovered a Lie bialgebra structure on the vector space generated by free homotopy classes of loops on an oriented surface. Goldman’s Lie bracket gives a lower bound on the minimum number of intersection points of two loops in two given free homotopy classes. Turaev’s Lie cobracket gives a lower bound on the minimum number of self-intersection points of a loop in a given free homotopy class. Chas showed that these bounds are not equalities in general. We show that for other operations, namely, the Andersen-Mattes-Reshetikhin Poisson bracket and a new operation μ, the corresponding bounds are always equalities. Some of this is joint work with Vladimir Chernov.

]]>Title: Finite subdivision rules and rational maps

Abstract: A finite subdivision rule gives an essentially combinatorial method for recursively subdividing planar complexes. The theory was developed (as part of an approach to Cannon’s conjecture) as a tool for studying the recursive structure at infinity of Gromov-hyperbolic groups, but it is becoming increasingly useful for studying postcritically finite rational maps. I’ll give an overview (with lots of graphic images) of some of the connections between finite subdivision rules and rational maps. ]]>

2pm:

Speaker Thomas Koberda

Title: The complex of curves for a right-angled Artin group

Abstract: I will discuss an analogue of the curve complex for right-angled Artin groups and describe some of its properties. I will then show how it guides parallel results between the theory of mapping class groups and the theory of right-angled Artin groups. Joint with Sang-hyun Kim.

3:30pm

Speaker: Eriko Hironaka

Title: Small dilatation pseudo-Anosov mapping classes

Abstract: A pseudo-Anosov mapping classes on a compact finite-type oriented surface S has the property that the growth rate of lengths of an essential simple closed curve under iterations of the mapping class is exponential, and the growth rate is independent of the choice of curve and the of the choice of metric. This growth rate is called that dilatation of the mapping class. In this talk, we discuss the problem of describing small dilatation pseudo-Anosov mapping classes, i.e., those such that the dilatation raised to the topological Euler characteristic of the surface is bounded. We describe small dilatation mapping classes in terms of deformations within fibered faces, and give some explicit examples. We finish the talk with a conjecture concerning the “shape” of small dilatation mapping classes.

5pm

Speaker: Andy Putman

Title: Stability in the homology of congruence subgroups

Abstract: I’ll discuss some recent results which uncover new patterns in the homology groups of congruence subgroups of SLn(ℤ) and related groups.