Past lectures

Abstract

Boundary theory provides a powerful tool for understanding „big groups“, i.e., the ones which admit a sufficiently rich behaviour at infinity (as, for instance, word hyperbolic groups or lattices in semi-simple Lie groups). We shall discuss interrelation between the geometrical and probabilistic approaches to the boundary theory and the ensuing applications to rigidity, geometric group theory and functional analysis.

The course will tentatively cover the following topics:

- Basic notions from measure theory, Lebesgue spaces, mesurable partitions

- Measure-theoretical boundaries of Markov chains

- Entropy of random walks

- The Poisson boundary of groups; criteria of triviality and maximality

- Relation with ergodic properties of geodesic and horocycle flows

- Amenability of groups and actions

- Compactifications of groups (end, visibility and hyperbolic compactifications)

- Boundary properties of semi-simple Lie groups, their discrete subgroups and symmetric spaces

- Boundary properties of the mapping class group and Teichmuller space