Analytic Group Theory

Summary of the program's focus area

Analytic group theory is an area focused on using analytic techniques for the study of infinite groups. The field can be thought of as beginning with work of Mostow, Furstenberg and Margulis on rigidity and super- rigidity for discrete subgroups of Lie groups, where their work used ideas from ergodic theory, functional analysis, probability, representation theory and quasi- conformal analysis. Soon after that work, mainly motivated by work of Alain Connes, similar ideas began to appear in the theory of operator algebras particularly when studying group and group action von Neumann algebras. A more geometric, but still analytic, set of approaches were spelled out in various works of Gromov, culminating in his book on Asymptotic Invariants of infinite groups. These three threads have been interwoven in much research on infinite groups over the past 40 years.

Over the last decade or so, these techniques have mixed with each other giving rise to an even richer theory, so much so that the scope has become rather large. We shall therefore focus on a few specific themes that add focus and gel with the theme in Section 5.1. We shall particularly be interested in analytic, ergodic theoretic and operator algebraic tools applied to geometric group theory. The following is a sample of sub-themes that are likely to be relevant during the semester.

(1) Measurable group theory and l2 invariants

(2) Amenability and amenable actions

(3) Property T for groups and von Neumann algebras 

(4) Commensurators

(5) Rigidity of groups, actions and operator algebras