We show that every graded nilpotent Lie group G of step r, equipped with a left invariant metric homogeneous with respect to the dilations induced by the grading, (this includes all Carnot groups ...with Carnot-Caratheodory metric) is Markov p-convex for all p∈2r,∞). We also show that this is sharp whenever G is a Carnot group with r≤3, a free Carnot group, or a jet space group; such groups are not Markov p-convex for any p∈(0,2r). This continues a line of research started by Li who proved this sharp result when G is the Heisenberg group. As corollaries, we obtain new estimates on the non-biLipschitz embeddability of some finitely generated nilpotent groups into nilpotent Lie groups of lower step. Sharp estimates of this type are known when the domain is the Heisenberg group and the target is a uniformly convex Banach space or L1, but not when the target is a nonabelian nilpotent group.
Abstract
We prove that if a simply connected nilpotent Lie group quasi-isometrically embeds into an $L^1$ space, then it is abelian. We reach this conclusion by proving that every Carnot group that ...bi-Lipschitz embeds into $L^1$ is abelian. Our proof follows the work of Cheeger and Kleiner, by considering the pull-back distance of a Lipschitz map into $L^1$ and representing it using a cut measure. We show that such cut measures, and the induced distances, can be blown up and the blown-up cut measure is supported on “generic” tangents of the original sets. By repeating such a blow-up procedure, one obtains a cut measure supported on half-spaces. This differentiation result then is used to prove that bi-Lipschitz embeddings can not exist in the non-abelian settings.
We prove that a self-homeomorphism of the Grushin plane is quasisymmetric if and only if it is metrically quasiconformal and if and only if it is geometrically quasiconformal. As the main step in our ...argument, we show that a quasisymmetric parametrization of the Grushin plane by the Euclidean plane must also be geometrically quasiconformal. We also discuss some aspects of the Euclidean theory of quasiconformal maps, such as absolute continuity on almost every compact curve, not satisfied in the Grushin case.
Stochastic embeddings of finite metric spaces into graph-theoretic trees have proven to be a vital tool for constructing approximation algorithms in theoretical computer science. In the present work, ...we build out some of the basic theory of stochastic embeddings in the infinite setting with an aim towards applications to Lipschitz free space theory. We prove that proper metric spaces stochastically embedding into \(\mathbb{R}\)-trees have Lipschitz free spaces isomorphic to \(L^1\)-spaces. We then undergo a systematic study of stochastic embeddability of Gromov hyperbolic metric spaces into \(\mathbb{R}\)-trees by way of stochastic embeddability of their boundaries into ultrametric spaces. The following are obtained as our main results: (1) Every snowflake of a compact, finite Nagata-dimensional metric space stochastically embeds into an ultrametric space and has Lipschitz free space isomorphic to \(\ell^1\). (2) The Lipschitz free space over hyperbolic \(n\)-space is isomorphic to the Lipschitz free space over Euclidean \(n\)-space. (3) Every infinite, finitely generated hyperbolic group stochastically embeds into an \(\mathbb{R}\)-tree, has Lipschitz free space isomorphic to \(\ell^1\), and admits a proper, uniformly Lipschitz affine action on \(\ell^1\).
We prove that the Lipschitz free space over a certain type of discrete metric space has the Radon-Nikodým property. We also show that the Lipschitz free space over a complete, locally compact metric ...space has the Schur or approximation property whenever the Lipschitz free space over each compact subset also has this property.
We show that every graded nilpotent Lie group \(G\) of step \(r\), equipped with a left invariant metric homogeneous with respect to the dilations induced by the grading, (this includes all Carnot ...groups with Carnot-Caratheodory metric) is Markov \(p\)-convex for all \(p \in 2r,\infty)\). We also show that this is sharp whenever \(G\) is a Carnot group with \(r \leq 3\), a free Carnot group, or a jet space 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. As corollaries, we obtain new estimates on the non-biLipschitz embeddability of some finitely generated nilpotent groups into nilpotent Lie groups of lower step. Sharp estimates of this type are known when the domain is the Heisenberg group and the target is a uniformly convex Banach space or \(L^1\), but not when the target is a nonabelian nilpotent group.
Given a finite collection \(\{X_i\}_{i\in I}\) of metric spaces, each of which has finite Nagata dimension and Lipschitz free space isomorphic to \(L^1\), we prove that their union has Lipschitz free ...space isomorphic to \(L^1\). The short proof we provide is based on the Pelczyński decomposition method. A corollary is a solution to a question of Kaufmann about the union of two planar curves with tangential intersection. A second focus of the paper is on a special case of this result that can be studied using geometric methods. That is, we prove that the Lipschitz free space of a union of finitely many quasiconformal trees is isomorphic to \(L^1\). These geometric methods also reveal that any metric quotient of a quasiconformal tree has Lipschitz free space isomorphic to \(L^1\). Finally, we analyze Lipschitz light maps on unions and metric quotients of quasiconformal trees in order to prove that the Lipschitz dimension of any such union or quotient is equal to 1.
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