Haagerup property and group-invariant percolation
Chiranjib Mukherjee, Konstantin Recke
TL;DR
This work provides probabilistic characterizations of the Haagerup property and Kazhdan's property (T) via Γ-invariant percolations on Cayley graphs, connecting geometric group properties to random subgraphs. It introduces a novel construction using invariant Poisson point processes on spaces with measured walls to generate percolations with large edge-marginals and quantitative two-point-function decay, yielding explicit exponential bounds in key Haagerup groups. The results extend to relative property (T) and have applications to Bernoulli percolation at the uniqueness threshold, while recovering and unifying aspects of the amenable case. Collectively, the paper answers longstanding questions of Lyons about probabilistic characterizations of nonamenable group properties and provides a versatile, quantitative framework for studying percolation on groups with Haagerup-type geometry.
Abstract
Let $\mathcal G$ be the Cayley graph of a finitely generated, infinite group $Γ$. We show that $Γ$ has the Haagerup property if and only if for every $α<1$, there is a $Γ$-invariant bond percolation $\mathbb P$ on $\mathcal G$ with $\mathbb E[\mathrm{deg}_ω(g)]>α\mathrm{deg}_{\mathcal G}(g)$ for every vertex $g$ and with the two-point function $τ(g,h)=\mathbb P\big[g\leftrightarrow h\big]$ vanishing as $d(g,h)\to\infty$. On the other hand, we show that $Γ$ has Kazhdan's property (T) if and only if there exists a threshold $α^*<1$ such that for every $Γ$-invariant bond percolation $\mathbb P$ on $\mathcal G$, $\mathbb E[\mathrm{deg}_ω(o)]>α^*\mathrm{deg}(o)$ implies that the two-point function is uniformly bounded away from zero. These results in particular answer questions raised by Lyons (J. Math. Phys. 41. 1099-1126 (2000)) about characterizations of properties of groups beyond amenability through group-invariant percolations. The method of proof is new and is based on a construction of percolations with suitable dependence structures built from invariant point processes on spaces with measured walls. This construction furthermore leads to quantitative bounds on the two-point functions, exhibiting in particular exponential decay of the two-point function in several prominent examples of Haagerup groups, including co-compact Fuchsian groups, co-compact discrete subgroups of $\mathrm{Isom}(\mathbb H^n)$ and lamplighters over free groups. This method also allows us to extend the aforementioned characterization of property (T) to the setting of relative property (T) and provide an application to Bernoulli percolation at the uniqueness threshold.