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Mathematica Slovaca

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Volume 68, Issue 2


Groups with positive rank gradient and their actions

Mark Shusterman
Published Online: 2018-03-31 | DOI: https://doi.org/10.1515/ms-2017-0106


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.

MSC 2010: 20E06; 20E07; 20E18; 20E26; 20F05; 20F65; 20F69; 43A05; 60B15; 58E40

Keywords: rank gradient; group actions; products of subgroups; Olshanskii’s theorem


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About the article

Received: 2016-02-09

Accepted: 2016-06-17

Published Online: 2018-03-31

Published in Print: 2018-04-25

Citation Information: Mathematica Slovaca, Volume 68, Issue 2, Pages 353–360, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.1515/ms-2017-0106.

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