Markov convexity and nonembeddability of the Heisenberg group

Li, Sean

Annales de l'Institut Fourier, Tome 66 (2016) no. 4, pp. 1615-1651.

Nous montrons que le groupe de Heisenberg ${ℍ}_{\infty }$ de dimension infinie est Markov 4-convexe et que le groupe de Heisenberg ${ℍ}_1$ de dimension 3 (et donc ${ℍ}_{\infty }$ 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 ${ℍ}_{\infty }$ ayant une distorsion d’au plus $C{n}^{1/4}\sqrt{logn}$. 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 ${ℍ}_{\infty }$, en montrant que la distorsion est de l’ordre de $\sqrt{logm}$.

We show that the continuous infinite dimensional Heisenberg group ${ℍ}_{\infty }$ is Markov 4-convex and that the 3-dimensional Heisenberg group ${ℍ}_1$ (and thus also ${ℍ}_{\infty }$) 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 ${ℍ}_{\infty }$ that has distortion at most $C{n}^{1/4}\sqrt{logn}$. 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 ${ℍ}_{\infty }$ by showing that the distortion is on the order of $\sqrt{logm}$.

Classification : 51F99
Keywords: Heisenberg group, Markov convexity, biLipschitz, embeddings
Mot clés : groupe de Heisenberg, convexité Markov, biLipschitz, plongements

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