Haagerup property and group-invariant percolation

Let G G be the Cayley graph of a finitely generated, infinite group Γ Γ . We show that Γ Γ has the Haagerup property if and only if for every α <1 α < 1 , there is a Γ Γ -invariant bond percolation P P on G G with E[ ω(g)]>α _G(g) E [ deg ω ( g ) ] > α deg G ( g ) for every vertex g and with the two-point function τ (g,h)=P [g↔ h ] τ ( g , h ) = P [ g ↔ h ] vanishing as d(g,h)→ ∞ d ( g , h ) → ∞ . As an upshot, we also obtain a sufficient condition for the Haagerup property using Bernoulli percolation and the associated threshold pconn p conn of random connected subgraphs of G G . On the other hand, we show that Γ Γ has Kazhdan’s property (T) if and only if there exists a threshold α ^*<1 α ∗ < 1 such that for every Γ Γ</mml