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

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 Matoušek 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.

This article supersedes arXiv:1411.3915 from the first author, 21 pages

References in corpus (1)

• Euclidean quotients of finite metric spaces

Cited by in corpus (3)

• On the geometry of the countably branching diamond graphs
• A metric interpretation of reflexivity for Banach spaces
• On Matoušek-like Embedding Obstructions of Countably Branching Graphs