This paper investigates the asymptotic behaviour of the minimal number of generators of finite index subgroups in residually finite groups. We analyze three natural classes of groups: amenable groups, groups possessing an infinite soluble normal subgroup and virtually free groups. As a tool for the amenable case we generalize Lackenby's trichotomy theorem on finitely presented groups.
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Miklós Abért, Andrei Jaikin-Zapirain, Nikolay Nikolov, The rank gradient from a combinatorial viewpoint. Groups Geom. Dyn. 5 (2011), no. 2, pp. 213–230DOI 10.4171/GGD/124