The ideal Poisson–Voronoi tessellation and some applications

We motivate and define the ideal Poisson–Voronoi tessellation (IPVT), a new random object with intriguing geometric properties when considered on a semisimple symmetric space (the hyperbolic plane, for example). In joint work with Mikolaj Frączyk and Sam Mellick, we use the IPVT to prove a result on the relationship between the volume of a manifold and the number of generators of its fundamental group (for higher rank semisimple Lie groups, the minimum number of generators in a lattice is sublinear in the covolume). In this talk we give some ideas of the proof. Time permitting, we will discuss related percolation results, joint with Matteo D'Achille, Jan Grebik, Ali Khezeli, and Konstantin Recke.