Poisson-Voronoi tessellations and fixed price in higher rank

TL;DR

This paper proves that lattices in higher rank semisimple Lie groups or products of automorphism groups of trees have fixed price one, introduces a novel Poisson-Voronoi tessellation method, and derives new homological growth estimates.

Contribution

It establishes fixed price one for lattices in higher rank groups using a novel Poisson-Voronoi tessellation approach, settling several conjectures.

Findings

Lattices in higher rank groups have cost one.

Minimal generators of torsion-free lattices grow sublinearly with co-volume.

New bounds on the growth of mod-$p$ homology groups in higher rank spaces.

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 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.