  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 $\sqrt{\log m}$.
