Geodesic
Rough similarity of left-invariant Riemannian metrics on some Lie groups
Enrico Le Donne1 orcid ; Gabriel Pallier2 orcid ; Xiangdong Xie3
1University of Fribourg, Fribourg, Switzerland; University of Jyväskylä, Jyväskylä, Finland
2Université de Lille, CNRS, UMR 8524, Lille, France
3Bowling Green State University, Bowling Green, USA
Groups, geometry, and dynamics, Tome 19 (2025) no. 1, pp. 227-263

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DOI

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.
DOI : 10.4171/ggd/790
Classification : 20F69, 22E25, 53C23
Mots-clés : Heintze groups, solvable groups, quasi-isometry, rough similarity

Enrico Le Donne  1   ; Gabriel Pallier  2   ; Xiangdong Xie  3

1 University of Fribourg, Fribourg, Switzerland; University of Jyväskylä, Jyväskylä, Finland
2 Université de Lille, CNRS, UMR 8524, Lille, France
3 Bowling Green State University, Bowling Green, USA 
Enrico Le Donne; Gabriel Pallier; Xiangdong Xie. Rough similarity of left-invariant Riemannian metrics on some Lie groups. Groups, geometry, and dynamics, Tome 19 (2025) no. 1, pp. 227-263. doi: 10.4171/ggd/790
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     title = {Rough similarity of left-invariant {Riemannian} metrics on~some {Lie} groups},
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     year = {2025},
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