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.
I would like to thank Alexander Olshanskii, Yves de Cornulier, Ashot Minasyan, Benjamin Steinberg, and Henry Wilton for many helpful discussions. This research was partially supported by a grant of the Israel Science Foundation with cooperation of UGC no. 40/14.
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