Simon Thomas (Rutgers University)

Asymptotic Cones of Finitely Generated Groups

In this series of talks, I will discuss one of the basic notions of Geometric Group Theory.  The talks are aimed at a general mathematical audience and there are essentially no prerequisites. Geometric Group Theory constitutes the third wave of combinatorial group theory.  In the first wave, combinatorial group theorists worked directly with words. Then came the realisation that more fun could be had and more progress made if they instead pretended to be doing something else.  In the second wave, they pretended to be doing very low dimensional topology.  In the third wave, following the example of Gromov, they are pretending to be doing geometry; i.e., they are regarding finitely generated groups as metric spaces and studying their ”large-scale'' or ”asymptotic'' geometries.  For example, if an observer moves steadily away from the Cayley graph of a finitely generated group, then any finite configuration will eventually become indistinguishable from a single point; but he may observe certain finite configurations which closely resemble earlier configurations.  The asymptotic cone is a topological space which encodes all of these recurring finite

configurations.  Unfortunately the actual construction of an asymptotic cone involves a number of non-canonical choices and it is not clear whether the resulting asymptotic cone depends on these choices.  Towards the end of this series of talks, answering a question of Gromov, I will present examples of finitely generated groups which have extremely non-homeomorphic asymptotic cones.

Anand Pillay  (University of Leeds)

Model theory, stability theory and the free group.

The series of talks will concern ramifications of Sela's important result that the first order theory Tfg of "the" noncommutative free group is stable.  The general thrust of the series will be: (i) "algebraic geometry over the free group" should really be the study of the category Def(Tfg) of definable sets in the theory Tfg, and (ii) the stability of Tfg provides a considerable amount of notions and machinery which can guide work on (i).

After introducing model theory and stability theory I hope to have time to discuss work on (i) and (ii) by my myself, Rizos Sklinos, as well as Chloe Perin.

Mark Feighn (Rutgers University)

An introduction to limit groups

What is the structure of Hom(G,F) where G is a finitely generated group and F is a free group? A class of groups called limit groups arises naturally in the study of this question.

We will give a series of introductory lectures on the subject of limit groups based on the joint paper with Mladen Bestvina `Notes on Sela’s work: Limit groups and Makanin-Razborov diagrams’. In particular, we will discuss an answer due to Kharlampovich-Myasnikov and Sela to the question above. Our focus will be on the more geometric approach of Sela.