Asymptotic dimension and uniform embeddings

We study uniform embeddings of metric spaces, which satisfy 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 the extra assumption that the space has the doubling property (finite Assouad dimension), that a space with polynomial growth has polynomial dimension growth and compression rate equal to one. The method used allows us to obtain a lower bound on the compression of the lamplighter group ℤ ≀ ℤ, which has infinite asymptotic dimension.