Isometries of nilpotent metric groups
[Isométries de groupes métriques nilpotents]
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.

Reçu le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/jep.48
Classification : 22E25,  53C30,  22F30
Mots clés : Isométries, groupes nilpotents, transformations affines, nilradical
@article{JEP_2017__4__473_0,
     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},
     volume = {4},
     year = {2017},
     doi = {10.5802/jep.48},
     mrnumber = {3646026},
     zbl = {1369.22006},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.48/}
}
Ville Kivioja; Enrico Le Donne. Isometries of nilpotent metric groups. Journal de l’École polytechnique — Mathématiques, Tome 4 (2017) , pp. 473-482. doi : 10.5802/jep.48. https://jep.centre-mersenne.org/articles/10.5802/jep.48/

[Are46] R. Arens - “Topologies for homeomorphism groups”, Amer. J. Math. 68 (1946), p. 593-610 | Article | MR 19916 | Zbl 0061.24306

[BB65] P. F. Baum & W. Browder - “The cohomology of quotients of classical groups”, Topology 3 (1965), p. 305-336 | Article | MR 189063 | Zbl 0152.22101

[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 | Article | MR 3553396 | Zbl 1370.53030

[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 | Article | MR 573347 | Zbl 0412.53026

[GW88] C. S. Gordon & E. N. Wilson - “Isometry groups of Riemannian solvmanifolds”, Trans. Amer. Math. Soc. 307 (1988) no. 1, p. 245-269 | Article | MR 936815 | Zbl 0664.53022

[Ham90] U. Hamenstädt - “Some regularity theorems for Carnot-Carathéodory metrics”, J. Differential Geom. 32 (1990) no. 3, p. 819-850 | Article | Zbl 0687.53041

[Hel01] S. Helgason - Differential geometry, Lie groups, and symmetric spaces, Graduate Texts in Math., vol. 34, American Mathematical Society, Providence, RI, 2001 | MR 1834454 | Zbl 0993.53002

[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 | Article | MR 804473 | Zbl 0581.55005

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

[Jab15a] M. Jablonski - “Homogeneous Ricci solitons”, J. reine angew. Math. 699 (2015), p. 159-182 | MR 3305924 | Zbl 1315.53046

[Jab15b] M. Jablonski - “Strongly solvable spaces”, Duke Math. J. 164 (2015) no. 2, p. 361-402 | Article | MR 3306558 | Zbl 1323.53049

[Kis03] I. Kishimoto - “Geodesics and isometries of Carnot groups”, J. Math. Kyoto Univ. 43 (2003) no. 3, p. 509-522 | Article | MR 2028665 | Zbl 1060.53039

[LN16] E. Le Donne & S. Nicolussi Golo - “Regularity properties of spheres in homogeneous groups”, Trans. Amer. Math. Soc. (2016) | Zbl 1385.53017

[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 | Article | MR 3441517 | Zbl 1343.53029

[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 | Article | MR 425012 | Zbl 0341.53030

[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 | Article | MR 1503467 | Zbl 65.1415.03

[MZ74] D. Montgomery & L. Zippin - Topological transformation groups, Robert E. Krieger Publishing Co., Huntington, N.Y., 1974 | Zbl 66.0959.03

[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 | Article | MR 412354 | Zbl 0318.53042

[Oze77] H. Ozeki - “On a transitive transformation group of a compact group manifold”, Osaka J. Math. 14 (1977) no. 3, p. 519-531 | MR 461377 | Zbl 0382.57020

[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 | Article | Zbl 0678.53042

[Sch68] H. Scheerer - “Homotopieäquivalente kompakte Liesche Gruppen”, Topology 7 (1968), p. 227-232 | Article | Zbl 0179.28404

[Sha04] Y. Shalom - “Harmonic analysis, cohomology, and the large-scale geometry of amenable groups”, Acta Math. 192 (2004) no. 2, p. 119-185 | Article | MR 2096453 | Zbl 1064.43004

[Wil82] E. N. Wilson - “Isometry groups on homogeneous nilmanifolds”, Geom. Dedicata 12 (1982) no. 3, p. 337-346 | Article | MR 661539 | Zbl 0489.53045

[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 | Article | MR 154228 | Zbl 0113.37101