Mathematics > Metric Geometry
[Submitted on 27 Apr 2014 (v1), last revised 31 Jan 2016 (this version, v3)]
Title:Markov convexity and nonembeddability of the Heisenberg group
View PDFAbstract:We compute the Markov convexity invariant of the continuous infinite dimensional Heisenberg group $\mathbb{H}_\infty$ to show that it is Markov 4-convex and cannot be Markov $p$-convex for any $p < 4$. As Markov convexity is a biLipschitz invariant and Hilbert space is Markov 2-convex, this gives a different proof of the classical theorem of Pansu and Semmes that the Heisenberg group does not admit a biLipschitz embedding into any Euclidean space.
The Markov convexity lower bound will follow from exhibiting an explicit embedding of Laakso graphs $G_n$ into $\mathbb{H}_\infty$ that has distortion at most $C n^{1/4} \sqrt{\log n}$. We use this to show that if $X$ is a Markov $p$-convex metric space, then balls of the discrete Heisenberg group $\mathbb{H}(\mathbb{Z})$ of radius $n$ embed into $X$ with distortion at least some constant multiple of
$$\frac{(\log n)^{\frac{1}{p}-\frac{1}{4}}}{\sqrt{\log \log n}}.$$
Finally, we show that Markov 4-convexity does not give the optimal distortion for embeddings of binary trees $B_m$ into $\mathbb{H}_\infty$ by showing that the distortion is on the order of $\sqrt{\log m}$.
Submission history
From: Sean Li [view email][v1] Sun, 27 Apr 2014 14:01:15 UTC (2,333 KB)
[v2] Mon, 7 Jul 2014 14:03:59 UTC (3,091 KB)
[v3] Sun, 31 Jan 2016 16:53:40 UTC (3,099 KB)
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