# Periodic points of algebraic actions of discrete groups

## TL;DR

This paper proves that Noetherian algebraic actions of finitely generated virtually nilpotent groups on compact abelian groups have a dense set of periodic points, advancing understanding of dynamical properties of such group actions.

## Contribution

It establishes the density of periodic points for Noetherian algebraic actions of finitely generated virtually nilpotent groups, a new result in the study of algebraic group actions.

## Key findings

- Noetherian actions have dense periodic points
- Finitely generated virtually nilpotent groups exhibit this property
- Advances understanding of algebraic dynamical systems

## Abstract

Let $\Gamma$ be a countable group. A $\Gamma$-action on a compact abelian group $X$ by continuous automorphisms of $X$ is called Noetherian if the dual of $X$ is Noetherian as a ${\mathbb Z}(\Gamma)$-module. We prove that any Noetherian action of a finitely generated virtually nilpotent group has a dense set of periodic points.