Amenable groups and measure concentration on spheres

Abstract

It is proved that a discrete group $G$ is amenable if and only if for every unitary representation of $G$ in an infinite-dimensional Hilbert space $\cal H$ the maximal uniform compactification of the unit sphere \s_{\cal H} has a $G$-fixed point, that is, the pair (\s_{\cal H},G) has the concentration property in the sense of Milman. Consequently, the maximal $U({\cal H})$-equivariant compactification of the sphere in a Hilbert space $\cal H$ has no fixed points, which answers a 1987 question by Milman. This is a version as of November 19, 1998, incorporating some revisions.Comment: 17 pages, LaTeX 2