Groups with positive rank gradient and their actions

We show that given a finitely generated LERF group G with positive rank gradient, and finitely generated subgroups A, B ≤ G of infinite index, one can find a finite index subgroup B₀ of B such that [G : 〈A ∪ B₀〉] = ∞. This generalizes a theorem of Olshanskii on free groups. We conclude that a finite product of finitely generated subgroups of infinite index does not cover G. We construct a transitive virtually faithful action of G such that the orbits of finitely generated subgroups of infinite index are finite. Some of the results extend to profinite groups with positive rank gradient.