Markov convexity and nonembeddability of the Heisenberg group
Annales de l'Institut Fourier, Volume 66 (2016) no. 4, pp. 1615-1651.

We show that the continuous infinite dimensional Heisenberg group is Markov 4-convex and that the 3-dimensional Heisenberg group 1 (and thus also ) cannot be Markov p-convex for any p<4. As Markov convexity is biLipschitz invariant and Hilbert spaces are Markov 2-convex, this gives a different proof of the classical theorem of Pansu and Semmes that the Heisenberg group does not biLipschitz embed into any Euclidean space.

The Markov convexity lower bound follows from exhibiting an explicit embedding of Laakso graphs G n into that has distortion at most Cn 1/4 logn. We use this to derive a quantitative lower bound for the biLipschitz distortion of balls of the discrete Heisenberg group into Markov p-convex metric spaces. Finally, we show surprisingly that Markov 4-convexity does not give the optimal distortion for embeddings of binary trees B m into by showing that the distortion is on the order of logm.

Nous montrons que le groupe de Heisenberg de dimension infinie est Markov 4-convexe et que le groupe de Heisenberg 1 de dimension 3 (et donc aussi) n’est pas Markov p-convexe pour tout p<4. Comme la convexité de Markov est un invariant bilipschitzien et les espaces de Hilbert sont Markov 2-convexes, on retrouve le théorème classique de Pansu et Semmes sur l’absence de plongement bilipschitzien du groupe de Heisenberg dans un espace euclidien.

La borne inférieure pour la convexité Markov suit de la construction d’un plongement de graphes de Laakso G n dans ayant une distorsion d’au plus Cn 1/4 logn. Nous obtenons ainsi une borne inférieure pour la distorsion bilipschitzienne des boules du groupe de Heisenberg discrète dans des espaces métriques Markov p-convexes. Enfin, nous montrons que, d’une manière surprenante, la 4-convexité de Markov ne donne pas la distorsion optimale pour les plongements d’arbres binaires B m en , en montrant que la distorsion est de l’ordre de logm.

Published online:
DOI: 10.5802/aif.3045
Classification: 51F99
Keywords: Heisenberg group, Markov convexity, biLipschitz, embeddings
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     title = {Markov convexity and nonembeddability of the {Heisenberg} group},
     journal = {Annales de l'Institut Fourier},
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Li, Sean. Markov convexity and nonembeddability of the Heisenberg group. Annales de l'Institut Fourier, Volume 66 (2016) no. 4, pp. 1615-1651. doi : 10.5802/aif.3045.

[1] Ball, Keith The Ribe programme, Astérisque (2013) no. 352, pp. Exp. No. 1047, viii, 147-159 (Séminaire Bourbaki. Vol. 2011/2012. Exposés 1043–1058)

[2] Bourgain, J. The metrical interpretation of superreflexivity in Banach spaces, Israel J. Math., Tome 56 (1986) no. 2, pp. 222-230 | DOI

[3] Cheeger, Jeff; Kleiner, Bruce Differentiating maps into L 1 , and the geometry of BV functions, Ann. of Math. (2), Tome 171 (2010) no. 2, pp. 1347-1385 | DOI

[4] Cheeger, Jeff; Kleiner, Bruce Metric differentiation, monotonicity and maps to L 1 , Invent. Math., Tome 182 (2010) no. 2, pp. 335-370 | DOI

[5] Cheeger, Jeff; Kleiner, Bruce; Naor, Assaf Compression bounds for Lipschitz maps from the Heisenberg group to L 1 , Acta Math., Tome 207 (2011) no. 2, pp. 291-373 | DOI

[6] Cygan, Jacek Subadditivity of homogeneous norms on certain nilpotent Lie groups, Proc. Amer. Math. Soc., Tome 83 (1981) no. 1, p. 69-70 | DOI

[7] Enflo, Per Banach spaces which can be given an equivalent uniformly convex norm, Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 1972), Tome 13 (1972), p. 281-288 (1973)

[8] James, Robert C. Uniformly non-square Banach spaces, Ann. of Math. (2), Tome 80 (1964), pp. 542-550

[9] James, Robert C. Super-reflexive Banach spaces, Canad. J. Math., Tome 24 (1972), pp. 896-904

[10] Johnson, William B.; Schechtman, Gideon Diamond graphs and super-reflexivity, J. Topol. Anal., Tome 1 (2009) no. 2, pp. 177-189 | DOI

[11] Laakso, Tomi J. Plane with A -weighted metric not bi-Lipschitz embeddable to N , Bull. London Math. Soc., Tome 34 (2002) no. 6, pp. 667-676 | DOI

[12] Lafforgue, Vincent; Naor, Assaf Vertical versus horizontal Poincaré inequalities on the Heisenberg group, Israel J. Math., Tome 203 (2014) no. 1, pp. 309-339 | DOI

[13] Lang, Urs; Plaut, Conrad Bilipschitz embeddings of metric spaces into space forms, Geom. Dedicata, Tome 87 (2001) no. 1-3, pp. 285-307 | DOI

[14] Lee, James R.; Naor, Assaf; Peres, Yuval Trees and Markov convexity, Geom. Funct. Anal., Tome 18 (2009) no. 5, pp. 1609-1659 | DOI

[15] Li, Sean Coarse differentiation and quantitative nonembeddability for Carnot groups, J. Funct. Anal., Tome 266 (2014) no. 7, pp. 4616-4704 | DOI

[16] Li, Sean; Raanan, Schul The traveling salesman problem in the Heisenberg group: upper bounding curvature (to appear in Trans. Amer. Math. Soc.)

[17] Li, Sean; Raanan, Schul An upper bound for the length of a traveling salesman path in the Heisenberg group (to appear in Rev. Mat. Iberoam.)

[18] Matoušek, Jiří On embedding trees into uniformly convex Banach spaces, Israel J. Math., Tome 114 (1999), pp. 221-237 | DOI

[19] Mendel, Manor; Naor, Assaf Markov convexity and local rigidity of distorted metrics, J. Eur. Math. Soc. (JEMS), Tome 15 (2013) no. 1, pp. 287-337 | DOI

[20] Montgomery, Richard A tour of subriemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs, Tome 91, American Mathematical Society, Providence, RI, 2002, xx+259 pages

[21] Naor, Assaf An introduction to the Ribe program, Jpn. J. Math., Tome 7 (2012) no. 2, pp. 167-233 | DOI

[22] Pansu, Pierre Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. (2), Tome 129 (1989) no. 1, pp. 1-60 | DOI

[23] Pisier, Gilles Martingales with values in uniformly convex spaces, Israel J. Math., Tome 20 (1975) no. 3-4, pp. 326-350

[24] Ribe, M. On uniformly homeomorphic normed spaces, Ark. Mat., Tome 14 (1976) no. 2, pp. 237-244

[25] Semmes, Stephen On the nonexistence of bi-Lipschitz parameterizations and geometric problems about A -weights, Rev. Mat. Iberoamericana, Tome 12 (1996) no. 2, pp. 337-410 | DOI

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