Summary: EXAMPLES OF RANDOM GROUPS
G. ARZHANTSEVA AND T. DELZANT
Abstract. We present Gromov's construction of a random group with no
coarse embedding into a Hilbert space.
In the late 1950's, working on the uniform classification of metric spaces,
Smirnov asked whether every separable metric space is uniformly homeomor-
phic to a subset of a Hilbert space [G]. This was settled negatively by Enflo,
who proved that the Banach space of null sequences c0 does not embed uniformly
homeomorphically into any Hilbert space [E].
Initiating a new theory, Gromov introduced the concept of a coarse embedding
(also termed as a uniform embedding) of metric spaces and asked whether every
separable metric space coarsely embeds into a Hilbert space [GrAI, p. 218].
Definition 1.1 (Coarse embedding). Let (X, d) be a metric space. Let H be a
separable Hilbert space. A map f : X H is said to be a coarse embedding if
for xn, yn X, n N,
d(xn, yn) if and only if f(xn) - f(yn) H .
Gromov's question was answered negatively in [Dr. et al.], where the authors
adapt Enflo's original construction and build an infinite family of finite graphs of
growing degrees admitting no coarse embedding into a Hilbert space.