Abstract
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 techniques, 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.
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Acknowledgements
This work was supported by a research Grant of the CAP (Comisión Académica de Posgrado) of the Universidad de la República. The authors are very grateful.
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Carrasco Piaggio, M., Sequeira, E. On quasi-isometry invariants associated to a Heintze group. Geom Dedicata 189, 1–16 (2017). https://doi.org/10.1007/s10711-016-0214-9
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DOI: https://doi.org/10.1007/s10711-016-0214-9