Groups, Geometry, and Dynamics
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Asymptotic dimension and uniform embeddings
Światosław R. Gal (1)
(1) Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384, WROCLAW, POLAND
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.
Keywords: Uniform embeddings, metric spaces, asymptotic dimension, Assouad–Nagata dimension, Hilbert space compression, coarse geometry