(beta)-distortion of some infinite graphs
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2016 London Mathematical Society. A distortion lower bound of ((h){1/p}) is proved for embedding the complete countably branching hyperbolic tree of height h into a Banach space admitting an equivalent norm satisfying property () of Rolewicz with modulus of power type p-in (1) (in short property (). Also it is shown that a distortion lower bound of ( {1/p}) is incurred when embedding the parasol graph with levels into a Banach space with an equivalent norm with property ( p). The tightness of the lower bound for trees is shown adjusting a construction of Matouek 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.