Let $G$ be an infinitely countable amenable group and let $K$ be a finite subset of $G$ containing the unit and at least one more element. A subset $C$ of $G$ is $K$-separated if the sets $Kc$ are disjoint as $c$ ranges over $C$. A $K$-separated set $C$ is maximal if no proper superset of $C$ is $K$-separated. The collection of the indicator functions of all maximal $K$-separated sets is closed and shift-invariant. We call it the $K$-shift. Last year, Benjy Weiss proposed to show that $K$-shifts are universal in the sense that any free ergodic measure-preserving action whose entropy is less than that of the $K$-shift is isomorphic to some invariant measure on the $K$-shift. Together with Benjy, Mateusz Więcek and Guohua Zhang we are trying to prove this. However an unexpected obstacle has been revealed (exactly two days ago) which makes the subject even more interesting.
Meeting ID: 852 4277 3200 Passcode: 103121