Embeddings of families of rescaled graphs into Cayley graphs,
examples of groups with exotic properties.

Cornelia Drutu

I shall explain two ways of embedding families of rescaled graphs into
Cayley graphs of groups. The first one allows to construct finitely
generated groups with continuously many non-homeomorphic asymptotic
cones (joint work with M. Sapir).  Note that by a result of Shelah,
Kramer, Tent and Thomas, under the Continuum Hypothesis, a finitely
generated group can have at most continuously many non-isometric
asymptotic cones.

The second way is less general, but it works for instance for families
of Cayley graphs of finite groups that are expanders. It allows to
construct finitely generated groups with (uniformly convex Banach
space)-compression taking any value in [0,1], and with asymptotic
dimension 2. In particular it gives the first example uniformly
embeddable in a Hilbert space with (uniformly convex Banach
space)-compression zero.  This is a joint work with G. Arzhantseva and
M.  Sapir.