On the sharpness of the Assouad embedding theorem for finitely generated groups of polynomial growth and nilpotent Lie groups

Abstract

Assouad's embedding theorem states that a proper snowflake of a doubling metric space embeds bilipschitzly into a Euclidean space. We investigate the sharpness of the bilipschitz distortion and target dimension of the Assouad embedding theorem for not virtually abelian finitely generated groups of polynomial growth with left-invariant word metric and nonabelian simply connected nilpotent Lie groups with left-invariant sub-Finsler Carnot-Carathéodory metric. We conclude that the distortion, namely $O(\varepsilon^{-1/2})$ for $(1-\varepsilon)$-snowflakes, cannot be improved, whereas the target dimension, namely $O(\varepsilon^{-O(1)})$, can be improved to $O(1)$ while maintaining distortion $O(\varepsilon^{-1/2})$. We establish the optimal target dimension in Chapter 1 by building on the approach of T. Tao (2020) for the Heisenberg group. Namely, we use a Nash--Moser iteration scheme along with an extension theorem for orthonormal vectorfields, the proof of which combines the Lovász local lemma and the concentration of measure phenomenon on the sphere and is the critical novelty of Chapter 1. We expect these embedding techniques to find use in metric dimension reduction. We establish the optimality of the distortion in Chapter 2 via `vertical versus horizontal inequalities,' which quantify the collapse of Lipschitz functions from the above groups into uniformly convex Banach spaces, which was qualitatively described by the Pansu--Semmes nonembeddability argument by Cheeger and Kleiner (2006) and Lee and Naor (2006). We prove the inequalities by using the vector-valued Littlewood--Paley--Stein theory approach of Lafforgue and Naor (2014) and by establishing a version of the classical Dorronsoro theorem inspired by Fässler and Orponen (2020). As a consequence, we compute the bilipschitz distortion of a ball of radius $n\ge 2$ in the above groups, where the Lie group metric is Riemannian, into Euclidean spaces of sufficiently large dimension or into $L^p$ ($1<p<\infty$) spaces to be $\Theta(\sqrt{\log n})$, and into $q(\ge 2)$-uniformly convex Banach spaces to be $\Omega((\log n)^{1/q})$.