BiLipschitz embeddings and nonembeddings of metric spaces and related problems

metric spaces;biLipschitz embeddings

Gartland, Christopher

关键词: metric spaces;    biLipschitz embeddings;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/107900/GARTLAND-DISSERTATION-2020.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

This thesis is concerned with problems relating to the Lipschitz category of metric spaces. We are chiefly interested in building machinery that can be used to deduce the existence or nonexistence of biLipschitz embeddings from one class of metric spaces into another. We will discuss two families of results along these lines.The first family deals with the problem of biLipschitz embeddability of metric spaces into Banach spaces with Radon-Nikod\'ym property (henceforth, RNP spaces). A major role in this story is played by differentiation theories of Lipschitz functions on metric measure spaces. Nonabelian Carnot groups are prime examples of spaces which support a good differentiation theory, and as a consequence they do not biLipschitz embed into any RNP space, as observed independently by Cheeger-Kleiner and Lee-Naor as a corollary of Pansu's theorem. In search of a nonlinear, metric characterization of the RNP, Ostrovskii found another class of metric spaces that do not biLipschitz embed into RNP spaces, namely spaces containing thick families of geodesics. His proof used an elementary martingale argument and involved no differentiation theory. Our first result is that any metric space containing a thick family of geodesics also contains a subset and a probability measure on that subset that supports a weak differentiation theory for RNP-valued Lipschitz functions. A corollary is a new nonembeddability result: the product of a Carnot group and an RNP space does not contain a biLipschitz copy of a thick family of geodesics. A second result from this project is that, if the metric space is a nonRNP Banach space, a subset consisting of a thick family of geodesics can be constructed to support a true differentiation theory of RNP-valued Lipschitz functions, like the one supported by Carnot groups. An intriguing question is whether the only obstructions to biLipschitz embeddability of complete metric spaces into RNP spaces, like the ones arising from differentiation theory, are local. If this question has a positive answer, it would imply that every complete, topologically discrete metric space biLipschitz embeds into an RNP space. Our third result is a proof of this statement in the special case where the metric space $(X,d)$ is \emph{essentially uniformly discrete}, meaning there is a $\theta > 0$ such that $|B_\theta(p)| < \infty$ for every $p \in X$. This generalizes a result of Kalton who proved that every uniformly discrete metric space biLipschitz embeds into an RNP space. Like Kalton, we prove our result by showing that the Lipschitz free space of $X$ has the RNP.The second family of results contained in this thesis is on the calculation of Markov convexity exponents of Carnot groups and applications. Markov convexity, developed by Lee-Naor-Peres and Mendel-Naor, is a biLipschitz and Lipschitz quotient invariant of metric spaces arising as a nonlinear generalization of the property of $p$-convexity of Banach spaces. It depends only on the finite subsets of the metric space and is thus of a different nature than theories of differentiation, which necessitate the existence of cluster points. Our first main result from this family is that every Carnot group $G$ of step $r$ is Markov $p$-convex for all $p \in [2r,\infty)$. Our second result is that this is sharp whenever $G$ is a Carnot group with $r \leq 3$ or a model filiform group; such groups are not Markov $p$-convex for any $p \in (0,2r)$. This continues a line of research started by Li who proved this sharp result when $G$ is the Heisenberg group. Finally, we obtain the following corollaries of these theorems, which are not attainable by differentiation methods: let $G$ be a Carnot group of step $r$ such that $r \leq 3$, $G$ is a free Carnot group, or $G$ is a jet space group. Let $G'$ be any Carnot group of step $r' < r$.\begin{enumerate}\item For any lattice $\Gamma \leq G$, the biLipschitz distortion of the $\Gamma$-ball of radius $R$ (with respect to a fixed finite generating set) into $G'$ is $\gtrsim \frac{\ln(R)^{\frac{1}{2r'}-\frac{1}{2r}}}{\ln(\ln(R))^{\frac{1}{2r'}+\frac{1}{2r}}}$.\item $G$ is not a Lipschitz quotient of any subset of $G'$.\item $G$ is not a Lipschitz quotient of any subset of $L^p$ (or any $p$-convex space) for any $p \in (1,2r)$.\item The model filiform group of infinite step is not a Lipschitz quotient of any subset of a superreflexive Banach space.\end{enumerate}The main question left open by this work is whether there is some Carnot group of step $r$ that is Markov $p$-convex for some $p < 2r$.