On quasi-isometry invariants associated to the derivation of a Heintze group

A a Heintze group is a Lie group of the form $N\rtimes_\alpha \mathbb{R}$, where $N$ is a simply connected nilpotent Lie group and $\alpha$ is a derivation of $\mathrm{Lie}(N)$ whose eigenvalues all have positive real parts. We show that if two purely real Heintze groups equipped with left-invariant metrics are quasi-isometric, then up to a positive scalar multiple, their respective derivations have the same characteristic polynomial. Using the same thecniques, we prove that if we restrict to the class of Heintze groups for which $N$ is the Heisenberg group, then the Jordan form of $\alpha$, up to positive scalar multiples, is a quasi-isometry invariant.