Uniform rank gradient, cost and local-global convergence

الأصل · EN

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. 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 with fixed price c, every Farber sequence has rank gradient 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.