# Nilpotent Cantor actions

## TL;DR

This paper characterizes nilpotent Cantor actions as unique among general Cantor actions through orbit equivalence and introduces new invariants for these actions.

## Contribution

It proves that any effective group action orbit equivalent to a nilpotent Cantor action is itself nilpotent, establishing a uniqueness property.

## Key findings

- Nilpotent Cantor actions are uniquely characterized by orbit equivalence.
- New invariants for nilpotent Cantor actions under continuous orbit equivalence.
- Effective actions orbit equivalent to nilpotent actions are themselves nilpotent.

## Abstract

A nilpotent Cantor action is a minimal equicontinuous action $\Phi \colon \Gamma \times \frak{X} \to \frak{X}$ on a Cantor set $\frak{X}$, where $\Gamma$ contains a finitely-generated nilpotent subgroup $\Gamma_0 \subset \Gamma$ of finite index. In this note, we show that these actions are distinguished among general Cantor actions: any effective action of a finitely generated group on a Cantor space, which is continuously orbit equivalent to a nilpotent Cantor action, must itself be a nilpotent Cantor action. As an application of this result, we obtain new invariants of nilpotent Cantor actions under continuous orbit equivalence.