JournalsggdVol. 19, No. 1pp. 227–263

Rough similarity of left-invariant Riemannian metrics on some Lie groups

  • Enrico Le Donne

    University of Fribourg, Fribourg, Switzerland; University of Jyväskylä, Jyväskylä, Finland

  • Gabriel Pallier

    Université de Lille, CNRS, UMR 8524, Lille, France

  • Xiangdong Xie

    Bowling Green State University, Bowling Green, USA
Rough similarity of left-invariant Riemannian metrics on some Lie groups cover
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Abstract

We consider Lie groups that are either Heintze groups or Sol-type groups, which generalize the three-dimensional Lie group SOL. We prove that all left-invariant Riemannian metrics on each such a Lie group are roughly similar via the identity map. This allows us to reformulate in a common framework former results by Le Donne–Xie, Eskin–Fisher–Whyte, Carrasco Piaggio, and recent results of Ferragut and Kleiner–Müller–Xie, on quasi-isometries of these solvable groups.

Cite this article

Enrico Le Donne, Gabriel Pallier, Xiangdong Xie, Rough similarity of left-invariant Riemannian metrics on some Lie groups. Groups Geom. Dyn. 19 (2025), no. 1, pp. 227–263

DOI 10.4171/GGD/790