This is an announcement for the paper "On the $(\beta)$-distortion of countably branching hyperbolic trees" by Florent Pierre Baudier.

Abstract: In this note we show that the distortion incurred by a bi-Lipschitz embedding of the countably branching hyperbolic tree of height $N$ into a Banach space admitting a norm satisfying Rolewicz property $(\beta)$ with power type $p>1$ is at least of the order of $\log(N)^{1/p}$. An application of our result gives a quantitative version of the non-embeddability of countably branching hyperbolic trees into reflexive Banach spaces admitting an equivalent asymptotically uniformly smooth norm and an equivalent asymptotically uniformly convex norm from Baudier, Kalton and Lancien.

Archive classification: math.MG math.FA

Mathematics Subject Classification: 46B20, 46B85

Remarks: 5 pages

Submitted from: florent@math.tamu.edu

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1411.3915

or

http://arXiv.org/abs/1411.3915