Ergodicity of algebraic actions of nilpotent groups

TL;DR

This paper proves that expansive algebraic actions of finitely generated nilpotent groups on connected compact abelian groups are ergodic, highlighting a specific class of group actions with this property.

Contribution

It establishes ergodicity for expansive algebraic actions of finitely generated nilpotent groups, and shows this does not extend to polycyclic groups.

Findings

Ergodicity holds for expansive algebraic actions of finitely generated nilpotent groups.

The result does not generalize to polycyclic groups.

Provides conditions under which algebraic group actions are ergodic.

Abstract

An algebraic $\Gamma$-action is an action of a countable group $\Gamma$ on a compact abelian group $X$ by continuous automorphisms of $X$. We prove that any expansive algebraic action of a finitely generated nilpotent group $\Gamma$ on a connected group $X$ is ergodic. We also show that this result does not hold for actions of polycyclic groups.