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

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A a Heintze group is a Lie group of the form NαR, where N is a simply connected nilpotent Lie group and α is a derivation of 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 α, up to positive scalar multiples, is a quasi-isometry invariant.