$(\beta)$-distortion of some infinite graphs

TL;DR

This paper establishes lower bounds on the distortion of embedding certain infinite graphs into Banach spaces with property (β), extending previous results and exploring implications for the structure and embeddability of Banach spaces.

Contribution

It provides new distortion lower bounds for embedding hyperbolic trees and parasol graphs into Banach spaces with property (β), and unifies various results on nonlinear stability and metric characterizations.

Findings

Lower bound of Ω(log(h)^{1/p}) for hyperbolic trees

Lower bound of Ω(ℓ^{1/p}) for parasol graphs

Extension of bounds to infinite trees and implications for Banach space theory

Abstract

A distortion lower bound of $\Omega(\log(h)^{1/p})$ is proven for embedding the complete countably branching hyperbolic tree of height $h$ into a Banach space admitting an equivalent norm satisfying property $(\beta)$ of Rolewicz with modulus of power type $p\in(1,\infty)$ (in short property ($\beta_p$)). Also it is shown that a distortion lower bound of $\Omega(\ell^{1/p})$ is incurred when embedding the parasol graph with $\ell$ levels into a Banach space with an equivalent norm with property ($\beta_p$). The tightness of the lower bound for trees is shown adjusting a construction of Matou\v{s}ek to the case of infinite trees. It is also explained how our work unifies and extends a series of results about the stability under nonlinear quotients of the asymptotic structure of infinite-dimensional Banach spaces. Finally two other applications regarding metric characterizations of asymptotic properties of Banach spaces, and the finite determinacy of bi-Lipschitz embeddability problems are discussed.