Lattice deformations in the Heisenberg group

The space of deformations of the integer Heisenberg group under the action of $\textrm{Aut}(H(\mathbb{R}))$ is a homogeneous space for a non-reductive group. We analyze its structure as a measurable dynamical system and obtain mean and variance estimates for Heisenberg lattice point counting in measurable subsets of $\mathbb{R}^3$; in particular, we obtain a random Minkowski-type theorem. Unlike the Euclidean case, we show there are necessary geometric conditions on the sets that satisfy effective variance bounds.