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 $\sqrt{ \log m}$.
Categories: math.MG math.FA