Uniform embeddability and exactness of free products

Abstract. Let A and B be countable discrete groups and let = A B be their free product. We show that if both A and B are uniformly embeddable in a Hilbert space then so is . We give two dierent proofs: the rst directly constructs a uniform embedding of from uniform embeddings of A and B; the second works without change to show that if both A and B are exact then so is . 1. Intoduction The concept of uniform embedding into Hilbert space was introduced by Gromov [Gro93]. It plays an important role in the study of the Novikov higher signature conjecture [FRR95, Yu00, STY00]. Let X be a countable discrete metric space and let d denote its metric; let H be a separable and innite-dimensional Hilbert space. A map F: X ! H is a uniform embedding [Gro93] if there exist non-decreasing functions 1 and 2 from R+ = [0;1) to R such that (1) 1(d(x; y)) k F (x) F (y) k 2(d(x; y)), for all x; y 2 X, and (2) limt!+1 i(t) = +1, for i = 1; 2. The discrete metric space X is uniformly embeddable if it admits exists a uniform embedding. A countable discrete group is exact if the functor given by the reduced crossed product with converts a short exact sequence -C-algebras into a short exact sequence of C-algebras. Equivalently, the functor given by spatial tensor product with the reduced C -algebra of converts one short exact sequence of C-algebras into another. The class of exact groups is closed under a number of operations [KW99], including the formation of free products (both with and without amalgam) [Dyk99, Dyk00]