On the H-type deviation of step two Carnot groups

The H-type deviation, $\delta({\mathbb G})$, of a step two Carnot group ${\mathbb G}$ quantifies the extent to which ${\mathbb G}$ deviates from the geometrically and algebraically tractable class of Heisenberg-type (H-type) groups. In an earlier paper, the author defined this notion and used it to provide new analytic characterizations for the class of H-type groups. In addition, a quantitative conjecture relating the H-type deviation to the behavior of the $\infty$-Laplacian of Folland's fundamental solution for the $2$-Laplacian was formulated; an affirmative answer to this conjecture would imply that all step two polarizable groups are of H-type. In this paper, we elucidate further properties of the H-type deviation. First, we show that $0\le \delta({\mathbb G}) \le 1$ for all step two Carnot groups ${\mathbb G}$. Recalling that $\delta({\mathbb F}_{2,m}) = \sqrt{(m-2)/m}$, where ${\mathbb F}_{2,m}$ is the free step two Carnot group of rank $m$, we conjecture that $\delta({\mathbb G}) \le \sqrt{(m-2)/m}$ for all step two rank $m$ groups. We explicitly compute $\delta({\mathbb G})$ when ${\mathbb G}$ is a product of Heisenberg groups and verify the conjectural upper bound for such groups, with equality if and only if ${\mathbb G}$ factors over the first Heisenberg group. We also prove the following rigidity statement: for each $m \ge 3$ there exists $\delta_0(m)>0$ so that if ${\mathbb G}$ is a step two and rank $m$ Carnot group with $\delta({\mathbb G}) < \delta_0(m)$, then ${\mathbb G}$ enjoys certain algebraic properties characteristic of H-type groups.