 
Summary: COARSE NONAMENABILITY AND COARSE EMBEDDINGS
GOULNARA ARZHANTSEVA, ERIK GUENTNER, AND J ´AN SPAKULA
ABSTRACT. We construct the first example of a coarsely nonamenable (= without
Guoliang Yu's property A) metric space with bounded geometry which coarsely em
beds into a Hilbert space.
1. INTRODUCTION
The purpose of this paper is to prove the following theorem:
Theorem 1.1. There exists a uniformly discrete metric space with bounded geometry,
which coarsely embeds into a Hilbert space, but does not have property A.
The concept of coarse embedding was introduced by Gromov [3, p. 218] in relation
to the Novikov conjecture (1965) on the homotopy invariance of higher signatures for
closed manifolds.
Definition 1.2. A metric space X is said to be coarsely embeddable into a Hilbert space
H if there exists a map f : X H such that for any xn,yn X, n N,
dist(xn,yn) f(xn) f(yn) H .
Yu established the coarse BaumConnes conjecture (1995) in topology for every
coarsely embeddable discrete space with bounded geometry [14, Theorem 1.1]. This
implies the Novikov conjecture for all closed manifolds whose fundamental group,
viewed with the word length metric, coarsely embeds into a Hilbert space. The re
sult confirmed Gromov's intuition and sparked an intense study of groups and metric
