BiLipschitz decomposition of Lipschitz maps between Carnot groups

Let $f : G \to H$ be a Lipschitz map between two Carnot groups. We show that if $B$ is ball of $G$, then there exists a subset $Z \subset B$, whose image in $H$ under $f$ has small Hausdorff content, such that $B \backslash Z$ can be decomposed into a controlled number of pieces, the restriction of $f$ on each of which is quantitatively biLipschitz. This extends a result of \cite{meyerson}, which proved the same result, but with the restriction that $G$ has an appropriate discretization. We provide an example of a Carnot group not admitting such a discretization.Comment: V2: 15 pages, added more background and details, slightly improved main theorem. Version to appear in Anal. Geom. Metr. Space

Markov convexity and nonembeddability of the Heisenberg group

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 $\sqrt{\log m}$.Comment: version to appear in Ann. Inst. Fourie

An upper bound for the length of a Traveling Salesman path in the Heisenberg group

We show that a sufficient condition for a subset $E$ in the Heisenberg group (endowed with the Carnot-Carath\'{e}odory metric) to be contained in a rectifiable curve is that it satisfies a modified analogue of Peter Jones's geometric lemma. Our estimates improve on those of \cite{FFP}, by replacing the power $2$ of the Jones-$\beta$-number with any power $r<4$. This complements (in an open ended way) our work \cite{Li-Schul-beta-leq-length}, where we showed that such an estimate was necessary, but with $r=4$.Comment: 19 pages. No figures; V2 several (inconsequential) errors corrected. V3 minor changes. Accepted to Revista Matem\'atica Iberoamerican