Mixing actions of Heisenberg group

TL;DR

This paper constructs various mixing actions of the Heisenberg group, including rank-one, Poisson, and Gaussian actions, and explores their properties, such as rigidity and restrictions to the center.

Contribution

It introduces new constructions of mixing actions of the Heisenberg group with specific properties, including rank-one, Poisson, Gaussian, and rigid weakly mixing actions.

Findings

Constructed mixing rank-one actions of $H_3(\mathbb{R})$

Developed mixing Poisson and Gaussian actions of $H_3(\mathbb{R})$

Created a rigid weakly mixing action with non-isomorphic restriction to the center

Abstract

Mixing (of all orders) rank-one actions $T$ of Heisenberg group $H_3(\Bbb R)$ are constructed. The restriction of $T$ to the center of $H_3(\Bbb R)$ is simple and commutes only with $T$. Mixing Poisson and mixing Gaussian actions of $H_3(\Bbb R)$ are also constructed. A rigid weakly mixing rank-one action $T$ is constructed such that the restriction of $T$ to the center of $H_3(\Bbb R)$ is not isomorphic to its inverse.