Sparse Unique Infinite Cluster Property for Graphs and Symmetric Spaces

We say that a group, graph, or more generally a locally compact metric space has the sparse unique infinite cluster (SUIC) property if it admits an automorphism-invariant random (closed) subset that is both connected and arbitrarily sparse. In their seminal paper, Hutchcroft and Pete showed that countable groups with property (T) have cost one by proving they have the SUIC property. Understanding when the SUIC property holds—or when it can be constructed by a randomized local algorithm (so-called factor of i.i.d.)—has turned out to be an interesting and difficult problem. Building on the recent breakthrough of Fraczyk, Mellick, and Wilkens, we propose a strategy for establishing the SUIC property using the well-studied Poisson–Voronoi percolation model. Specifically, we show that for higher-rank symmetric spaces with property (T), the Poisson–Voronoi percolation model has vanishing uniqueness threshold as the intensity tends to zero. In this talk, I will introduce the relevant concepts and outline the key ideas connecting percolation theory and geometry. This is joint work with Konstantin Recke.