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
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
Every notion in the talk will be carefully explained.
You can watch the lecture here: