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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Uniform rank gradient, cost, and local-global convergence


Authors: Miklós Abért and László Márton Tóth
Journal: Trans. Amer. Math. Soc. 373 (2020), 2311-2329
MSC (2010): Primary 05C99, 20E15, 20E26, 20F69, 37A15
DOI: https://doi.org/10.1090/tran/8008
Published electronically: January 7, 2020
MathSciNet review: 4069220
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Abstract:

We analyze the rank gradient of finitely generated groups with respect to sequences of subgroups of finite index that do not necessarily form a chain, by connecting it to the cost of p.m.p. (probability measure preserving) actions. We generalize several results that were only known for chains before. The connection is made by the notion of local-global convergence.

In particular, we show that for a finitely generated group $\Gamma$ with fixed price $c$, every Farber sequence has rank gradient $c-1$. By adapting Lackenby’s trichotomy theorem to this setting, we also show that in a finitely presented amenable group, every sequence of subgroups with index tending to infinity has vanishing rank gradient.


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Additional Information

Miklós Abért
Affiliation: MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary
Email: abert.miklos@renyi.mta.hu

László Márton Tóth
Affiliation: Central European University, Budapest, Hungary; and MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary
ORCID: 0000-0002-6821-8060
Email: toth.laszlo.marton@renyi.mta.hu

Received by editor(s): September 4, 2018
Received by editor(s) in revised form: March 12, 2019
Published electronically: January 7, 2020
Additional Notes: The authors were supported by the Hungarian National Research, Development and Innovation Office, NKFIH grant K109684 and the ERC Consolidator Grant 648017.
Article copyright: © Copyright 2019 American Mathematical Society