Isometries of nilpotent metric groups

Kivioja, Ville; Le Donne, Enrico

Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 473-482.

Nous considérons des groupes de Lie munis de distances arbitraires. Nous supposons seulement que ces distances sont invariantes à gauche et induisent la topologie de la variété sous-jacente. Nous appelons groupes de Lie métriques de tel objets. Mis à part les groupes de Lie riemanniens, des exemples remarquables sont donnés par les groupes de Lie sous-riemanniens, les groupes homogènes et, en particulier, les groupes de Carnot munis de distances de Carnot–Carathéodory. Nous montrons la régularité des isométries, c’est-à-dire des homéomorphismes qui préservent la distance. Notre premier résultat est l’analyticité de telles applications entre des groupes de Lie métriques. Le second résultat est que, si deux groupes de Lie métriques sont connexes et nilpotents, alors toute isométrie entre ces groupes est la composition d’une translation à gauche et d’un isomorphisme. Il y a des contre-exemples si on ne suppose pas que les groupes sont connexes ou nilpotents. Le premier résultat repose sur la solution du cinquième problème de Hilbert par Montgomery et Zippin. Le second résultat est démontré à l’aide du premier, en réduisant le problème au cas riemannien, cas qui a été essentiellement résolu par Wolf.

We consider Lie groups equipped with arbitrary distances. We only assume that the distances are left-invariant and induce the manifold topology. For brevity, we call such objects metric Lie groups. Apart from Riemannian Lie groups, distinguished examples are sub-Riemannian Lie groups, homogeneous groups, and, in particular, Carnot groups equipped with Carnot–Carathéodory distances. We study the regularity of isometries, i.e., distance-preserving homeomorphisms. Our first result is the analyticity of such maps between metric Lie groups. The second result is that if two metric Lie groups are connected and nilpotent then every isometry between the groups is the composition of a left translation and an isomorphism. There are counterexamples if one does not assume the groups to be either connected or nilpotent. The first result is based on a solution of the Hilbert’s fifth problem by Montgomery and Zippin. The second result is proved, via the first result, reducing the problem to the Riemannian case, which was essentially solved by Wolf.

Classification : 22E25, 53C30, 22F30
Keywords: Isometries, nilpotent groups, affine transformations, nilradical
Mot clés : Isométries, groupes nilpotents, transformations affines, nilradical

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