This is an announcement for the paper "Markov convexity and nonembeddability of the Heisenberg group" by Sean Li.

Abstract: We compute the Markov convexity invariant of the continuous Heisenberg group $\mathbb{H}$ 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 $\mathbb{H}$ 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 somewhat unexpectedly that the optimal distortion of embeddings of binary trees $B_m$ into the infinite dimensional Heisenberg group is on the order of $\sqrt{\log m}$

Archive classification: math.MG math.FA

Remarks: 20 pages

Submitted from: seanli@math.uchicago.edu

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1404.6751

or

http://arXiv.org/abs/1404.6751