Vadim Kaimanovich (Université de Ottawa) : "Freedom and boundaries."

When is the group action on the Poisson boundary of a random walk free? If not, what can one say about the point stabilizers?

The assumption that a measure class preserving action of the group of integers is free does not really reduce the generality and is routinely made in ergodic theory. This is apparently the reason why actions of bigger countable groups with non-trivial point stabilizers haven't really been considered until about 15 years ago when Vershik introduced the class of totally non-free actions (the ones for which all point stabilizers are different). Equivalently, an action is totally non-free if it can be realized as the action by conjugation on the space of subgroups. Invariant measures of the latter action are currently quite popular under the probabilistically flavoured  name of IRS (invariant random subgroups).

I will outline recent results (joint with Anna Erschler) on the stabilizers of the group action on its Poisson boundary: existence of a free boundary action for any group with infinite conjugacy classes, a complete description of the possible kernels of such actions, and an example of a totally non-free boundary action.