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Groups with positive rank gradient and their actions

Mark Shusterman
From the journal Mathematica Slovaca


We show that given a finitely generated LERF group G with positive rank gradient, and finitely generated subgroups A, BG of infinite index, one can find a finite index subgroup B0 of B such that [G : 〈AB0〉] = ∞. 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.


Communicated by Denis Osin


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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Received: 2016-2-9
Accepted: 2016-6-17
Published Online: 2018-3-31
Published in Print: 2018-4-25

© 2018 Mathematical Institute Slovak Academy of Sciences