Expansiveness of vertical subgroups of the Heisenberg group

TL;DR

This paper investigates the properties of expansive subsets within the Heisenberg group, focusing on vertical subgroups and their limitations in the context of group actions on compact metric spaces.

Contribution

It extends the concept of expansiveness from abelian groups to the non-abelian Heisenberg group, analyzing vertical subgroups and their expansiveness properties.

Findings

The center of the Heisenberg group cannot be expansive when acting on an infinite space.

There always exists at least one nonexpansive 2D-dimensional vertical subgroup.

The study adapts ideas from Boyle and Lind for $bZ^D$ actions to the Heisenberg group.

Abstract

In the paper we study expansiveness along distinguished subsets in the case of a continuous action of the discrete Heisenberg group on a compact metric space $(\mathbb X,\rho)$. Transferring the ideas proposed by Boyle and Lind for continuous actions of $\mathbb{Z}^D$, we embed the acting group in the (continuous) $(2D+1)$-dimensional Heisenberg group $\mathcal H$ and define expansive subsets of $\mathcal H$. We focus on the expansiveness of vertical subgroups of the Heisenberg group. In particular, we show that, if only the space $\mathbb X$ is infinite, the center of $\mathcal H$ cannot be expansive, and that there always exists at least one nonexpansive $2D$-dimensional vertical subgroup.