Lewis Bowen: "Entropy, orbit-equivalence and free groups"
Abstract: We would like to classify actions of groups up to measure-conjugacy (MC) and orbit-equivalence (OE). There are also quantitative restrictions on orbit-equivalence, for example, bounded orbit-equivalence (BOE) and integrable orbit-equivalence (IOE) which lie between MC and OE. Entropy is an MC invariant and one of the primary tools in classifying actions up to MC. However, it is not an OE-invariant. Tim Austin proved that it is an IOE-invariant for actions of amenable groups and this result has been extended by Kerr-Li to groups containing a weakly normal amenable subgroup. I will discuss the first result of this kind for actions of non-abelian free groups: f-entropy is BOE-invariant. No knowledge of the classical entropy theory or the f-invariant will be assumed.