6,694 research outputs found
BiLipschitz decomposition of Lipschitz maps between Carnot groups
Let be a Lipschitz map between two Carnot groups. We show that
if is ball of , then there exists a subset , whose image in
under has small Hausdorff content, such that can be
decomposed into a controlled number of pieces, the restriction of on each
of which is quantitatively biLipschitz. This extends a result of
\cite{meyerson}, which proved the same result, but with the restriction that
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 to show that it is Markov
4-convex and cannot be Markov -convex for any . 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 into that has distortion
at most . We use this to show that if is a Markov
-convex metric space, then balls of the discrete Heisenberg group
of radius embed into with distortion at least
some constant multiple of
Finally, we show that Markov 4-convexity does not give the optimal distortion
for embeddings of binary trees into by showing that
the distortion is on the order of .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 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 of the Jones--number with any power . 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 .Comment: 19 pages. No figures; V2 several (inconsequential) errors corrected.
V3 minor changes. Accepted to Revista Matem\'atica Iberoamerican
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