Kolmogorov's theorem for sofic groups

Speaker: Miklós Abert

Abstract: Let Gamma be a countable group. A Bernoulli process on Gamma is defined by tossing a random coin independently for each element of Gamma. An old question of von Neumann asks when two Bernoulli processes are isomorphic, as invariant random colorings.

In 1959, Sinai and Kolmogorov proved that if they are, then the base random coins have the same entropy. They did this by introducing an entropy notion for such processes, starting a large field of research. The result has been later generalized by Ornstein and Weiss to all amenable groups. In the same paper they also presented a paradox for the free group, similar and connected to the Banach-Tarski paradox. This result convinced everyone for decades that there is no meaningful entropy theory for non-amenable groups.

It turned out that everyone was wrong, when Bowen recently generalized Kolmogorov's theorem for sofic groups in a breakthrough result. This class in particular includes free groups and as of now, no one knows a group that is not sofic.

Recently, with Weiss, we introduced a new entropy notion for processes on sofic groups. Among other things, this leads to a quite transparent proof of Bowen's theorem, that I hope to present fully at the talk.

Every notion in the talk will be carefully explained.