Coarse embeddings of locally finite metric spaces into Banach spaces without cotype

M. Gromov [7] suggested to use coarse embeddings into a Hilbert space or into a uniformly convex space as a tool for solving some of the well-known problems. G. Yu [21] and G. Kasparov and G. Yu [11] have shown that this is indeed a very powerful tool. On the other hand, there exist separable metric spaces ([6] and [5, Section 6]) which are not coarsely embeddable into Hilbert spaces. In [9] (see, also, [8] and [20]) it was shown that such spaces exist even among Cayley graphs of finitely presented groups. Nonembeddability results in the Banach space theory setting were obtained in [10], [19], and [15]. In [10] it was shown that `p (p > 2) in not coarsely embeddable into `2, in [19] this result was strengthened to a characterization of quasi-Banach spaces which are coarsely embeddable into a Hilbert space. In [15] it was shown that cotype is an obstruction for coarse embeddability of Banach spaces. See [14], [18] and [17] for more non-embeddability results. Going in another direction, one can try to find spaces, maybe not as good as uniformly convex spaces, such that every finitely presented group, or, more generally, every metric space with bounded geometry is coarsely embeddable into it. (Recall that a metric space A is said to have a bounded geometry if for each r > 0 there exist a positive integer M(r) such that each ball in A of radius r contains at most M(r) elements.) Each separable