Accessible 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.

References

[1] Abert, M.—Gelander, T.—Nikolov, N.: Rank, combinatorial cost and homology torsion growth in higher rank lattices, (2015), preprint. Search in Google Scholar

[2] Abert, M.—Jaikin-Zapirain, A.—Nikolov, N.: The rank gradient from a combinatorial viewpoint, Groups Geom. Dyn. 5 (2011), 213–230. Search in Google Scholar

[3] Abert, M.—Nikolov, N.: Rank gradient, cost of groups and the rank versus Heegaard genus problem, J. Eur. Math. Soc. 14 (2012), 1657–1677.10.4171/JEMS/344 Search in Google Scholar

[4] Chaynikov, V.: Actions of maximal growth of hyperbolic groups, (2012), preprint. Search in Google Scholar

[5] de Cornulier, Y.: Finitely presented wreath products and double coset decompositions, Geom. Dedic. 122 (2006), 89–108.10.1007/s10711-006-9061-4 Search in Google Scholar

[6] Kar, A.—Nikolov, N.: Cost and rank gradient of Artin groups and their relatives, Groups Geom. Dyn. 8 (2014), 1195–1205.10.4171/GGD/300 Search in Google Scholar

[7] Lackenby, M.: Expanders, rank and graphs of groups, Israel J. Math. 146 (2005), 357–370.10.1007/BF02773541 Search in Google Scholar

[8] Minasyan, A.: Some properties of subsets of hyperbolic groups, Comm. Algebra 33 (2005), 909–935.10.1081/AGB-200051164 Search in Google Scholar

[9] Olshanskii, A.: On pairs of finitely generated subgroups in free groups, Proc. Amer. Math. Soc. 143 (2015), 4177–4188.10.1090/proc/12537 Search in Google Scholar

[10] Osin, D.: Rank gradient and torsion groups, Bull. Lond. Math. Soc. 43 (2011), 10–16.10.1112/blms/bdq075 Search in Google Scholar

[11] Pappas, N.: Rank gradient and p-gradient of amalgamated free products and HNN extensions, Comm. Algebra 43 (2015), 4515–4527.10.1080/00927872.2014.948631 Search in Google Scholar

[12] Schlage-Puchta, J. C.: A p-group with positive Rank Gradient, J. Group Theory 15 (2012), 261–270. Search in Google Scholar

[13] Shusterman, M.: Ranks of subgroups in boundedly generated groups, Bull. Lond. Math. Soc. 48 (2016), 539–547.10.1112/blms/bdw023 Search in Google Scholar

[14] Shusterman, M.: Ascending chains of finitely generated subgroups, (2016), preprint. Search in Google Scholar

[15] Snopce, I.—Zalesskii, P.: Subgroup properties of pro-p extensions of centralisers, Sel. Math. New Ser. 20 2014, 465–489.10.1007/s00029-013-0128-4 Search in Google Scholar

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