Uniformly Bounded Representations and Exact Groups

We characterize groups with Guoliang Yu’s property A (i.e., exact groups) by the existence of a family of uniformly bounded representations which approximate the trivial representation. Property A is a large scale geometric property that can be viewed as a weak counterpart of amenability. It was shown in [12], that for a finitely generated group property A implies the Novikov conjecture. It was also quickly realized that this notion has many other applications and interesting connections, see [9,10]. A well-known characterization of amenability states that the constant function 1 on G, as a coefficient of the trivial representation, can be approximated by diagonal, finitely supported coefficients of the left regular representation of G on `2(G). In this note we prove a counterpart of this result for groups with property A in terms of uniformly bounded representations. A representation π of a group G on a Hilbert space H is said to be uniformly bounded if supg∈G ‖πg‖B(H) <∞. Theorem 1. Let G be a finitely generated group equipped with a word length function. G has property A (i.e., G is exact) if and only if for every ε> 0 there exists a uniformly bounded representation π of G on a Hilbert space H, a vector v ∈ H and a constant S > 0 such that (1) ‖πgv‖ = 1 for all g ∈G, (2) |1−〈πgv,πhv〉| ≤ ε if |g−1h| ≤ 1, (3) 〈πgv,πhv〉 = 0 if |g−1h| ≥ S. Alternatively, the second condition can be replaced by an almost-invariance condition: ‖πgv−πhv‖ ≤ ε if |g−1h| ≤ 1. Another characterization of property A in this spirit, involving convergence for isometric representations on Hilbert C∗-modules was studied in [4]. Recall that the Fell topology on the unitary dual is defined using convergence of coefficients of unitary representations. Theorem 1 states that the trivial representation can be approximated by uniformly bounded representations, in a fashion similar to Fell’s topology. Similar phenomena were considered by M. Cowling [2,3] in the case of the Lie group Sp(n,1). Recall that Sp(n,1) has property (T), and thus the trivial representation is an isolated point among the equivalence classes of unitary Date: 22 April 2013.