Abstract
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 equation for this class of groups. We prove the minimal number of generators of a torsion-free lattice in $G$ is sublinear in its co-volume, settling a conjecture of Abért–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 rank locally symmetric spaces. Our method of proof is novel, using low intensity Poisson point processes on $G$ and the geometry of their associated Voronoi tesselations. We prove, as the intensity limits to zero, these tessellations partition the space into “horoball-like” cells such 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.
