Canal-U

Point processes, cost and the growth of rank for locally compact groups (workshop ERC Nemo Processus ponctuels et graphes aléatoires unimodulaires)

The cost of a vertex transitive graph is the infimum of the expected degree of an invariant random wiring of the graph. Similarly, one can define the cost of a point process on a homogeneous space, as the infimum of the average degree of a factor wiring on its points. It turns out that the cost of a Poisson process is maximal among point processes of the same density, by proving that all free processes weakly contain the Poisson. The cost is related to the growth of the minimal number of generators of lattices in Lie groups. We expect that for semisimple Lie groups, the minimal number of generators is sublinear in the volume except for SL(2,R). We outline partial results in this direction and pose some open problems. One of them is to compute the cost of the Poisson on hyperbolic 3-space: solving this would lead to the solution of a 40 year old problem on Heegaard genus of hyperbolic 3-manifolds. This is joint work with Samuel Mellick.