Published online by Cambridge University Press: 21 January 2013
Mixing (of all orders) rank-one actions $T$ of the Heisenberg group
${H}_{3} ( \mathbb{R} )$ are constructed. The restriction of
$T$ to the center of
${H}_{3} ( \mathbb{R} )$ is simple and commutes only with
$T$. Mixing Poisson and mixing Gaussian actions of
${H}_{3} ( \mathbb{R} )$ are also constructed. A rigid weakly mixing rank-one action
$T$ is constructed such that the restriction of
$T$ to the center of
${H}_{3} ( \mathbb{R} )$ is not isomorphic to its inverse.