Haagerup property and group-invariant percolation

Abstract Let $${\mathcal {G}}$$ G be the Cayley graph of a finitely generated, infinite group $$\Gamma $$ Γ . We show that $$\Gamma $$ Γ has the Haagerup property if and only if for every $$\alpha \alpha \deg _{{\mathcal {G}}}(g)$$ E [ deg ω ( g ) ] > α deg G ( g ) for every vertex g and with the two-point function $$\tau (g,h)=\mathbb {P}\big [g\leftrightarrow h\big ]$$ τ ( g , h ) = P [ g ↔ h ] vanishing as $$d(g,h)\rightarrow \infty $$ d ( g , h ) → ∞ . As an upshot, we also obtain a sufficient condition for the Haagerup property using Bernoulli percolation and the associated threshold $$p_{\textrm{conn}}$$ p conn of random connected subgraphs of $$\mathcal G$$ G . On the other hand, we show that $$\Gamma $$ Γ has Kazhdan’s property (T) if and only if there exists a threshold $$\alpha ^*