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

TL;DR

This paper investigates invariants of Heintze groups under quasi-isometries, showing that the characteristic polynomial and Jordan form of the derivation are preserved up to scalar multiples, thus advancing understanding of their geometric classification.

Contribution

It establishes that the characteristic polynomial and Jordan form of the derivation are quasi-isometry invariants for certain classes of Heintze groups, providing new tools for their geometric analysis.

Findings

Characteristic polynomial is a quasi-isometry invariant for purely real Heintze groups.

Jordan form of the derivation is invariant for Heintze groups with Heisenberg nilpotent part.

Results connect algebraic properties of derivations to geometric quasi-isometry classifications.

Abstract

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.