Uniform rank gradient, cost and local-global convergence

TL;DR

This paper explores the relationship between rank gradient, cost, and local-global convergence in finitely generated groups, extending previous results to more general subgroup sequences and establishing new connections.

Contribution

It generalizes the concept of rank gradient to non-chain subgroup sequences and links it to cost via local-global convergence, broadening the scope of prior work.

Findings

Farber sequences in groups with fixed price have rank gradient equal to cost minus one.

In finitely presented amenable groups, subgroup sequences with increasing index have zero rank gradient.

The results extend known theorems to more general subgroup sequences beyond chains.

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.