Random Walks on Groups, Commutative and Non-commutative Dynamics

Introduction

The general goal of the proposed research is to gain a deeper understanding of group properties which are reflected by the theory of random walks. Another goal is to reveal further connections between this theory and the rigidity phenomenon. The main mathematical fields appearing in this research plan are measurable and topological group actions (Ergodic Theory and Topological Dynamics), and group actions on C*-algebras.

Objectives

Connes’ Rigidity Conjecture

One of the main objectives is developing a theory towards solving a specific case of Connes’ Rigidity Conjecture, formulated for C*-algebras. Namely, differentiating reduced C*-algebras of irreducible lattices of different ranks.

The suggested approach is inspired by a well-known rigidity result of Furstenberg. This involves studying the relationship between measurable and topological boundaries, as well as their C*- and von Neumann algebraic counterparts. Related to this relationship, it is also conjectured that:

1. The existence of uniquely ergodic models for probability measure preserving actions in a much wider setup than is currently known.

Automorphism Groups of Markov Chains

Another goal is to develop a theory of automorphism groups of Markov chains. Two potential applications are discussed:

1. Developing new techniques for realizing the Furstenberg-Poisson boundary.
2. Relating the boundaries of groups, which are measure equivalent.

Additional Research Directions

An additional line of research suggests new systematic studies of operator algebras related to groups. This direction is inspired by the fruitful theme in Geometric Group Theory, studying the space of all subgroups of a given group.

The dynamics on the space of subalgebras is suggested to provide a new set of invariants attributed to groups, unitary representations, and group actions. A subalgebra rigidity phenomenon is conjectured to hold for higher rank groups, and a strategy based on Boundary Theory is being presented.

This direction opens many new horizons to the study of groups’ operator algebras.