Salomon Bochner Lectures in Mathematics, 2003-2004

Whilst geometric group theory has become a very classical topic during the two last decades, its relative, Measure Equivalence, is a recent focus of interest. Geometric group theory proposes to consider abstract groups as geometric objects, and to study algebraic properties from the point of view of large-scale geometry. Measure equivalence, on the other hand, aims at studying abstract groups through ergodic-theoretic techniques. We will give an introduction to this programme.

The structure of (measurable, countable) equivalence relations on probability spaces has certain analogies with the theory of factors in operator algebras. It is typically difficult to say much about the "quotient" structure, that is, about the orbit space. It turn out that there is a direct connection between these questions and measure equivalence; we will present recent results in this direction, using geometric tools.

We will present the basic notions of negative curvature for general metric spaces: Gromov-hyperbolicity, CAT(-1) geometry. The boundary at infinity of such spaces is particularly useful for the understanding of groups of isometries. On the other hand, we will consider random walks and explain how the associated Poisson boundary relates to the boundary of the metric space.

We will give a short and application-oriented introduction to bounded cohomology, focusing on the tools that can be used in rigidity questions.

We will present a cohomological viewpoint for group actions on negatively curved spaces. Amongst the applications, we will (1) prove superrigidity statements for actions of lattices on such spaces; (2) indicate how to use this viewpoint for the ergodic-theoretical questions addressed in the first lectures.