Ergodic Ramsey Theory for Non-amenable Groups
Time and place
12:30 PM on Tuesday, April 4th, 2017; NAC 6/310
Prof. Hillel Furstenberg (Hebrew University of Jerusalem)
Abstract
Ramsey theory treats the phenomenon that certain patterns must appear inside sufficiently large subsets of structures of a certain type. The prototype is the assertion conjectured by Erdos and Turan and proved by Szemeredi, that a subset of the integers with positive density possesses arbitrarily long arithmetic progressions. This has an ergodic-theoretic proof based on recurrence patterns that show up in any ergodic dynamical system. One can ask for similar phenomena for "large" subsets of other groups. It is not difficult to generalize the "dynamic" approach to amenable groups because of the existence of an invariant density notion. We show that for arbitrary group actions there is a notion of measures invariant "on the average". With this we can define the analogue of subsets of positive density and prove a general version of the following: If a subset of a finitely generated free group has the property that for every sufficiently large L, it contains a fixed positive proportion of all words of length L, then it contains geometric progressions of every length.
