On the disjointness property of groups and a conjecture of Furstenberg

Eli Glasner, Benjamin Weiss

Published 2018-07-23Version 1

In his seminal 1967 paper "Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation" Furstenberg introduced the notion of disjointness of dynamical systems, both topological and measure preserving. In this paper he showed that for actions of the integers the Bernoulli system $\Omega = \{0, 1\}^\mathbb{Z}$, is disjoint from every minimal system, and that the subring $R_0$, over the field $\mathbb{Z}_2 =\{0, 1\}$, generated by the minimal functions in $\Omega$, is a proper subset of $\Omega$. He conjectured that a similar result holds in general and in a 1983 article we confirmed this by showing that the closed subalgebra $\mathfrak{A}$ of $l^\infty(\mathbb{Z})$, generated by the minimal functions, is a proper ubalgebra of $l^\infty(\mathbb{Z})$. In this work we generalize these results to a large class of groups. We call a countable group $G$ a DJ group if for every metrizable minimal action of $G$ there exists an essentially free minimal action disjoint from it. We extend the above results to all DJ groups. Amenable and residually finite groups are DJ, and the DJ property extends through short exact sequences. We define a simple dynamical condition DDJ on minimal systems and we say that a group $G$ is DDJ if every metrizable minimal $G$-system has this property. The DJ property implies DDJ and by means of an intricate construction we show that every finitely generated DDJ group is also DJ.