Expansiveness of vertical subgroups of the Heisenberg group

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.