The nonexistence of expansive polycyclic group actions on the circle $\mathbb S^1$

TL;DR

This paper proves that no expansive actions by polycyclic groups can exist on the circle, demonstrating a fundamental limitation on group actions in this setting.

Contribution

It establishes the nonexistence of expansive polycyclic group actions on the circle, a new result in the study of group actions on one-dimensional manifolds.

Findings

No expansive polycyclic group actions on the circle exist.

Provides a classification constraint for group actions on $ ext{S}^1$.

Enhances understanding of dynamical restrictions for group actions.

Abstract

We show that the circle $\mathbb S^1$ admits no expansive polycyclic group actions.