Poisson-Voronoi percolation in higher rank

We show that the uniqueness thresholds for Poisson-Voronoi percolation in symmetric spaces of connected higher rank semisimple Lie groups with property (T) converge to zero in the low-intensity limit. This phenomenon is fundamentally different from situations in which Poisson-Voronoi percolation has previously been studied. Our approach builds on a recent breakthrough of Fraczyk, Mellick and Wilkens (arXiv:2307.01194) and provides an alternative proof strategy for Gaboriau's fixed price problem. As a further application of our result, we give a new class of examples of non-amenable Cayley graphs that admit factor of iid bond percolations with a unique infinite cluster and arbitrarily small expected degree, answering a question inspired by Hutchcroft-Pete (Invent. math. 221 (2020)).}