Estimates on the Markov Convexity of Carnot Groups and Quantitative Nonembeddability

TL;DR

This paper investigates the Markov convexity properties of Carnot groups and graded nilpotent Lie groups, establishing sharp bounds that influence their embeddability into lower-step nilpotent groups and extending previous results from the Heisenberg group.

Contribution

It provides new estimates on the Markov p-convexity of Carnot groups, identifying sharp thresholds and implications for nonembeddability into lower-step nilpotent groups.

Findings

Carnot groups are Markov p-convex for p ≥ 2r

Sharp non-convexity results for certain Carnot groups when p < 2r

Implications for non-biLipschitz embeddability of nilpotent groups

Abstract

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.