We show that for any Carnot group
there exists a natural number
such that for any
the
metric space
admits a
bi-Lipschitz embedding into
with distortion
.
We do this by building on the approach of T. Tao (Rev. Mat.Iberoam. 37:1 (2021), 1–44), who established the above assertion when
is the
Heisenberg group using a new variant of the Nash–Moser iteration scheme combined
with a new extension theorem for orthonormal vector fields. Beyond the need
to overcome several technical issues that arise in the more general setting
of Carnot groups, a key point where our proof departs from that of Tao
is in the proof of the orthonormal vector field extension theorem, where
we incorporate the Lovász local lemma and the concentration of measure
phenomenon on the sphere in place of Tao’s use of a quantitative homotopy
argument.
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