Title: 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. 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 $\Gamma$ 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.