L6

I will consider automorphism groups of countable structures acting continuously on compact spaces: the viewpoint of topological dynamics. A beautiful paper of Kechris, Pestov and Todorcevic makes a connection between this and the ‘structural Ramsey theory’ of Nesetril, Rodl and others in finite combinatorics. I will describe some results and questions in the area and say how the Hrushovski predimension constructions provide answers to some of these questions (but then raise more questions). This is joint work with Hubicka and Nesetril.

An abelian l-group G is essentially a partially ordered subgroup of functions from a set to a totally ordered abelian group such

that G is closed under taking finite infima and suprema. For example, G could be the continuous semi-linear functions defined on the open
unit square, or, G could be the continuous semi-algebraic functions defined in the plane with values in (0,\infty), where the group
operation is multiplication. I will show how G, under natural geometric assumptions, can be interpreted (in a weak sense) in its lattice of
zero sets. This will then be applied to the model theory of natural divisible abelian l-groups. For example we will see that the
aforementioned examples are elementary equivalent. (Parts of the results have been announced in a preliminary report from 1987 by F. Shen
and V. Weispfenning.)

Given a collection F of holomorphic functions, we consider how to describe all the holomorphic functions locally definable from F. The notion of local definability of holomorphic functions was introduced by Wilkie, who gave a complete description of all functions locally definable from F in the neighbourhood of a generic point. We prove that this description is not complete anymore in the neighbourhood of non-generic points. More precisely, we produce three examples of holomorphic functions which each suggest that at least three new definable operations need to be added to Wilkie's description in order to capture local definability in its entirety. The construction illustrates the interaction between resolution of singularities and definability in the o-minimal setting. Joint work with O. Le Gal, G. Jones, J. Kirby.

I am going to discuss Rips' conjecture that all finitely presented groups with quadratic Dehn functions have decidable conjugacy problem.

This is a joint work with A.Yu. Olshanskii.
 

It is known that if the boundary of a 1-ended
hyperbolic group G has a local cut point then G splits over a 2-ended group. We prove a similar theorem for CAT(0)
groups, namely that if a finite set of points separates the boundary of a 1-ended CAT(0) group G
then G splits over a 2-ended group. Along the way we prove two results of independent interest: we show that continua separated
by finite sets of points admit a tree-like decomposition and we show a splitting theorem for nesting actions on R-trees.
This is joint work with Eric Swenson.

Finiteness properties of groups come in many flavours, I will discuss topological finiteness properties. These relate to the finiteness of skelata in a classifying space. Groups with interesting finiteness properties have been found in many ways, however all such examples contains free abelian subgroups of high rank. I will discuss some constructions of groups discussing the various ways we can reduce the rank of a free abelian subgroup. 

In this talk I'll give a general presentation of my recent work that a purely loxodromic Kleinian group of Hausdorff dimension<1 is a classical Schottky group. This gives a complete classification of all Kleinian groups of dimension<1. The proof uses my earlier result on the classification of Kleinian groups of sufficiently small Hausdorff dimension. This result in conjunction to another work (joint with Anderson) provides a resolution to Bers uniformization conjecture. No prior knowledge on the subject is assumed.

Torsion in homology are invariants that have received increasing attention over the last twenty years, by the work of Lück, Bergeron, Venkatesh and others. While there are various vanishing results, no one has found a finitely presented group where the torsion in the first homology is exponential over a normal chain with trivial intersection. On the other hand, conjecturally, every 3-manifold group should be an example.

A group is right angled if it can be generated by a list of infinite order elements, such that every element commutes with its neighbors. Many lattices in higher rank Lie groups (like SL(n,Z), n>2) are right angled. We prove that for a right angled group, the torsion in the first homology has subexponential growth for any Farber sequence of subgroups, in particular, any chain of normal subgroups with trivial intersection. We also exhibit right angled cocompact lattices in SL(n,R) (n>2), for which the Congruence Subgroup Property is not known. This is joint work with Nik Nikolov and Tsachik Gelander.