Poisson-Voronoi tessellations in higher rank and the fixed price conjecture

The cost of a probability measure preserving action of a countable group G on X is an invariant that generalizes the rank (minimal number of generators) of G and measures the “minimal average number of maps” needed to connect every pair of points of X in the same G orbit. The fixed price conjecture predicts that any two essentially free p.m.p. actions of the group G have the same cost. I will talk about a recent joint work with Sam Mellick and Amanda Wilkens in which we prove fixed price one for higher rank lattices in semisimple real Lie groups. As a corollary, we show that the number of generators or the dimension of the first mod-p homoloogy of index n subgroup of such a group grows like o(n). The proof is based on certain miraculous properties of Poisson-Voronoi tessellations of higher rank symmetric spaces that might be of independent interest.