From homogeneous metric spaces to Lie groups

Cowling, Michael G.; Kivioja, Ville; Le Donne, Enrico; Nicolussi Golo, Sebastiano; Ottazzi, Alessandro

Comptes Rendus. Mathématique, Tome 362 (2024) no. G9, pp. 943-1014

We study homogeneous metric spaces, by which we mean connected, locally compact metric spaces whose isometry group acts transitively.

After a review of a number of classical results, we use the Gleason–Iwasawa–Montgomery–Yamabe–Zippin structure theory to show that for all positive $ϵ$, each such space is $(1,ϵ)$-quasi-isometric to a connected metric Lie group (metrized with a left-invariant distance that is not necessarily Riemannian).

Next, we develop the structure theory of Lie groups to show that every homogeneous metric manifold is homeomorphically roughly isometric to a quotient space of a connected amenable Lie group, and roughly isometric to a simply connected solvable metric Lie group.

Third, we investigate solvable metric Lie groups in more detail, and expound on and extend work of Gordon and Wilson [31, 32] and Jablonski [44] on these, showing, for instance, that connected solvable Lie groups may be made isometric if and only if they have the same real-shadow.

Finally, we show that homogeneous metric spaces that admit a metric dilation are all metric Lie groups with an automorphic dilation.

Nous étudions les espaces métriques homogènes, c’est-à-dire, les espaces métriques connexes et localement compacts dont le groupe d’isométries agit transitivement.

Après avoir passé en revue un certain nombre de résultats classiques, nous utilisons la théorie de la structure de Gleason–Iwasawa–Montgomery–Yamabe–Zippin dans le but de montrer que pour tout $ϵ$ positif, chacun des espaces susmentionnés est $(1,ϵ)$-quasi-isométrique à un groupe de Lie métrique connexe (métrizé par une distance invariante à gauche non nécessairement riemannienne).

Ensuite, nous développons la théorie de la structure des groupes de Lie pour montrer que toute variété métrique homogène est grossièrement isométrique par homéomorphisme au quotient d’un groupe de Lie moyennable, connexe, et grossièrement isométrique à un groupe de Lie métrique résoluble simplement connexe.

Troisièmement, nous étudions plus en détail les groupes de Lie métriques résolubles, et nous développons et étendons les travaux de Gordon et Wilson [31, 32] et de Jablonski [44] sur ceux-ci, en montrant, par exemple, que les groupes de Lie résolubles connexes peuvent être rendus isométriques si et seulement s’ils ont la même ombre réelle.

Enfin, nous montrons que les espaces métriques homogènes qui admettent une homothétie métrique sont tous des groupes de Lie métriques possédant une homothétie automorphe.

Classification : 53C30, 22F30, 20F69, 22E25
Keywords: Homogeneous spaces, Structure, Lie groups
Mots-clés : Espaces homogènes, structure, groupes de Lie

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