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Abstract

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 l^p-) 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.
The method allows to obtain the lower bound of the compression of the
lamplighter group ZwrZ, which has infinite asymptotic dimension.