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Licensed Unlicensed Requires Authentication Published by De Gruyter March 31, 2018

Groups with positive rank gradient and their actions

Mark Shusterman
From the journal Mathematica Slovaca

Abstract

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.


URL: markshus.wix.com/math

Communicated by Denis Osin


Acknowledgement

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

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