Title: The Shannon-McMillan-Breiman theorem beyond amenable groups,
Abstract: We introduce a new isomorphism-invariant notion of entropy for measure
preserving actions of arbitrary countable groups on probability spaces, which
we call cocycle entropy. We develop methods to show that cocycle entropy
satisfies many of the properties of classical amenable entropy theory, but
applies in much greater generality to actions of non-amenable groups. One key
ingredient in our approach is a proof of a subadditive convergence principle
which is valid for measure-preserving amenable equivalence relations, going
beyond the Ornstein-Weiss Lemma for amenable groups.
For a large class of countable groups, which may in fact include all of them,
we prove the Shannon-McMillan-Breiman pointwise convergence theorem for cocycle
entropy in their measure-preserving actions.
We also compare cocycle entropy to Rokhlin entropy, and using an important
recent result of Seward we show that they coincide for free, ergodic actions of
any countable group in the class. Finally, we use the example of the free group
to demonstrate the geometric significance of the entropy equipartition property
implied by the Shannon-McMillan-Breiman theorem.