Uniform rank gradient, cost, and local-global convergence

Abstract

We analyze the rank gradient of finitely generated groups with respect to sequences of subgroups of finite index that do not necessarily form a chain, by connecting it to the cost of p.m.p. (probability measure preserving) actions. We generalize several results that were only known for chains before. The connection is made by the notion of local-global convergence.

In particular, we show that for a finitely generated group Γ \Gamma with fixed price c c , every Farber sequence has rank gradient c − 1 c-1 . By adapting Lackenby’s trichotomy theorem to this setting, we also show that in a finitely presented amenable group, every sequence of subgroups with index tending to infinity has vanishing rank gradient.