Swiatoslaw R. Gal (Uniwersytet Wroclawski)
Asymptotic dimension and uniform embeddings
We study uniform embeddings of metric spaces satisfying some asymptotic
tameness conditions such as finite asymptotic dimension, finite Assouad-Nagata
dimension, polynomial dimension growth or polynomial growth into function
spaces. We show how the type function of a space with finite asymptotic
dimension estimates its Hilbert (or any lp-) compression. In particular, we
show that the spaces of finite asymptotic dimension with linear type (spaces
with finite Assouad-Nagata dimension) have compression rate equal to one. We
show, without an extra assumption that the space has doubling property (finite
Assouad dimension), that a space with polynomial growth has polynomial
dimension growth and compression rate equal to one.