Poisson-Voronoi tessellations and fixed price in higher rank

Let $G$ be a higher rank semisimple real Lie group or the product of at least two automorphism groups of regular trees. We prove all probability measure preserving actions of lattices in such groups have cost one, answering Gaboriau's fixed price question for this class of groups. We prove the minimal number generators of a torsion-free lattice in $G$ is sublinear in the co-volume of $\Gamma$, settling a conjecture of Ab\'{e}rt-Gelander-Nikolov. As a consequence, we derive new estimates on the growth of first mod-$p$ homology groups of higher rank locally symmetric spaces. Our method of proof is novel, using low intensity Poisson point processes on higher rank symmetric spaces and the geometry of their associated Voronoi tessellations. We prove as the intensities limit to zero, these tessellations partition the space into ``horoball-like'' cells so that any two share an unbounded border. We use this new phenomenon to construct low cost graphings for orbit equivalence relations of higher rank lattices.