Hyperbolic groups admit proper affine isometric actions on l p - spaces Guoliang

Let X be a Banach space and Γ be a countable discrete group. An affine and isometric action α of Γ on X is said to be proper if limg→∞‖α(g)ξ‖ = ∞ for every ξ ∈ X. If Γ admits a proper isometric affine action on Hilbert space, then Γ is said to be of Haagerup property [9] or a-T-menable [12]. Bekka, Cherix and Valette proved that an amenable group admits a proper affine isometric action on Hilbert space [3]. This result has important applications to K-theory of group C-algebras [13] [14]. It is well known that an infinite Property (T) group doesn’t admit a proper affine isometric action on Hilbert space. The purpose of this paper is to prove the following result.