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

TL;DR

This paper establishes lower bounds on the distortion when embedding certain infinite graphs into Banach spaces with Rolewicz property $(eta)$, revealing limitations based on graph height and structure.

Contribution

It provides new quantitative lower bounds on graph embeddings into Banach spaces with Rolewicz property $(eta)$, advancing understanding of geometric embedding constraints.

Findings

Lower bound of $oldsymbol{ ext{Omega}( ext{log}(h)^{1/p})}$ for hyperbolic trees

Lower bound of $oldsymbol{ ext{Omega}(l^{1/p})}$ for parasol graphs

Discussion on the optimality and applications of these bounds

Abstract

We show a distortion lower bound of $\Omega(\log(h)^{1/p})$ when embedding the countably branching hyperbolic tree of height $h$ into a Banach space with an equivalent norm satisfying Rolewicz property $(\beta)$ with modulus of power type $p>1$. Similarly we show that a distortion lower bound of $\Omega(l^{1/p})$ is incurred when embedding the parasol graphs with $l$ levels into a Banach space with the above property. We discuss the optimality of our results as well as several applications.