Isometries of nilpotent metric groups
Journal de l’École polytechnique — Mathématiques, Volume 4 (2017), pp. 473-482.

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.

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.

Published online:
DOI: 10.5802/jep.48
Classification: 22E25, 53C30, 22F30
Keywords: Isometries, nilpotent groups, affine transformations, nilradical
Mot clés : Isométries, groupes nilpotents, transformations affines, nilradical
Ville Kivioja 1; Enrico Le Donne 1

1 Department of Mathematics and Statistics, University of Jyväskylä 40014 Jyväskylä, Finland
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
     author = {Ville Kivioja and Enrico Le Donne},
     title = {Isometries of nilpotent metric groups},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {473--482},
     publisher = {\'Ecole polytechnique},
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Ville Kivioja; Enrico Le Donne. Isometries of nilpotent metric groups. Journal de l’École polytechnique — Mathématiques, Volume 4 (2017), pp. 473-482. doi : 10.5802/jep.48.

[Are46] R. Arens - “Topologies for homeomorphism groups”, Amer. J. Math. 68 (1946), p. 593-610 | DOI | MR | Zbl

[BB65] P. F. Baum & W. Browder - “The cohomology of quotients of classical groups”, Topology 3 (1965), p. 305-336 | DOI | MR | Zbl

[CKL + ] M. Cowling, V. Kivioja, E. Le Donne, E., S. Nicolussi Golo & A. Ottazzi - “From homogeneous metric spaces to Lie groups”, in preparation

[CL16] L. Capogna & E. Le Donne - “Smoothness of subRiemannian isometries”, Amer. J. Math. 138 (2016) no. 5, p. 1439-1454 | DOI | MR | Zbl

[Cor15] Y. Cornulier - “On the quasi-isometric classification of locally compact groups” (2015), arXiv:1212.2229

[Gor80] C. Gordon - “Riemannian isometry groups containing transitive reductive subgroups”, Math. Ann. 248 (1980) no. 2, p. 185-192 | DOI | MR | Zbl

[GW88] C. S. Gordon & E. N. Wilson - “Isometry groups of Riemannian solvmanifolds”, Trans. Amer. Math. Soc. 307 (1988) no. 1, p. 245-269 | DOI | MR | Zbl

[Ham90] U. Hamenstädt - “Some regularity theorems for Carnot-Carathéodory metrics”, J. Differential Geom. 32 (1990) no. 3, p. 819-850 | DOI | Zbl

[Hel01] S. Helgason - Differential geometry, Lie groups, and symmetric spaces, Graduate Texts in Math., vol. 34, American Mathematical Society, Providence, RI, 2001 | MR | Zbl

[HK85] J. R. Hubbuck & R. M. Kane - “The homotopy types of compact Lie groups”, Israel J. Math. 51 (1985) no. 1-2, p. 20-26 | DOI | MR | Zbl

[HN12] J. Hilgert & K.-H. Neeb - Structure and geometry of Lie groups, Springer Monographs in Mathematics, Springer, New York, 2012 | DOI

[Jab15a] M. Jablonski - “Homogeneous Ricci solitons”, J. reine angew. Math. 699 (2015), p. 159-182 | MR | Zbl

[Jab15b] M. Jablonski - “Strongly solvable spaces”, Duke Math. J. 164 (2015) no. 2, p. 361-402 | DOI | MR | Zbl

[Kis03] I. Kishimoto - “Geodesics and isometries of Carnot groups”, J. Math. Kyoto Univ. 43 (2003) no. 3, p. 509-522 | DOI | MR | Zbl

[LN16] E. Le Donne & S. Nicolussi Golo - “Regularity properties of spheres in homogeneous groups”, Trans. Amer. Math. Soc. (2016) | Zbl

[LO16] E. Le Donne & A. Ottazzi - “Isometries of Carnot Groups and Sub-Finsler Homogeneous Manifolds”, J. Geom. Anal. 26 (2016) no. 1, p. 330-345 | DOI | MR | Zbl

[LR17] E. Le Donne & S. Rigot - “Besicovitch Covering Property on graded groups and applications to measure differentiation”, J. reine angew. Math. (2017)

[Mil76] J. Milnor - “Curvatures of left invariant metrics on Lie groups”, Adv. in Math. 21 (1976) no. 3, p. 293-329 | DOI | MR | Zbl

[MS39] S. B. Myers & N. E. Steenrod - “The group of isometries of a Riemannian manifold”, Ann. of Math. (2) 40 (1939) no. 2, p. 400-416 | DOI | MR | Zbl

[MZ74] D. Montgomery & L. Zippin - Topological transformation groups, Robert E. Krieger Publishing Co., Huntington, N.Y., 1974 | Zbl

[OT76] T. Ochiai & T. Takahashi - “The group of isometries of a left invariant Riemannian metric on a Lie group”, Math. Ann. 223 (1976) no. 1, p. 91-96 | DOI | MR | Zbl

[Oze77] H. Ozeki - “On a transitive transformation group of a compact group manifold”, Osaka J. Math. 14 (1977) no. 3, p. 519-531 | MR | Zbl

[Pan89] P. Pansu - “Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un”, Ann. of Math. (2) 129 (1989) no. 1, p. 1-60 | DOI | Zbl

[Sch68] H. Scheerer - “Homotopieäquivalente kompakte Liesche Gruppen”, Topology 7 (1968), p. 227-232 | DOI | Zbl

[Sha04] Y. Shalom - “Harmonic analysis, cohomology, and the large-scale geometry of amenable groups”, Acta Math. 192 (2004) no. 2, p. 119-185 | DOI | MR | Zbl

[Wil82] E. N. Wilson - “Isometry groups on homogeneous nilmanifolds”, Geom. Dedicata 12 (1982) no. 3, p. 337-346 | DOI | MR | Zbl

[Wol63] J. A. Wolf - “On locally symmetric spaces of non-negative curvature and certain other locally homogeneous spaces”, Comment. Math. Helv. 37 (1962/1963), p. 266-295 | DOI | MR | Zbl

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