## Uniform rank gradient, cost, and local-global convergence

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- by Miklós Abért and László Márton Tóth PDF
- Trans. Amer. Math. Soc.
**373**(2020), 2311-2329 Request permission

## 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.

## References

- Miklós Abért and Gábor Elek,
*Dynamical properties of profinite actions*, Ergodic Theory Dynam. Systems**32**(2012), no. 6, 1805–1835. MR**2995875**, DOI 10.1017/S0143385711000654 - M. Abért and G. Elek,
*The Space of Actions, Partition Metric and Combinatorial Rigidity*, preprint, arxiv:1108.2147. - Miklós Abért and Nikolay Nikolov,
*Rank gradient, cost of groups and the rank versus Heegaard genus problem*, J. Eur. Math. Soc. (JEMS)**14**(2012), no. 5, 1657–1677. MR**2966663**, DOI 10.4171/JEMS/344 - Miklós Abért, Andrei Jaikin-Zapirain, and Nikolay Nikolov,
*The rank gradient from a combinatorial viewpoint*, Groups Geom. Dyn.**5**(2011), no. 2, 213–230. MR**2782170**, DOI 10.4171/GGD/124 - Miklos Abert, Tsachik Gelander, and Nikolay Nikolov,
*Rank, combinatorial cost, and homology torsion growth in higher rank lattices*, Duke Math. J.**166**(2017), no. 15, 2925–2964. MR**3712168**, DOI 10.1215/00127094-2017-0020 - Miklós Abért and Benjamin Weiss,
*Bernoulli actions are weakly contained in any free action*, Ergodic Theory Dynam. Systems**33**(2013), no. 2, 323–333. MR**3035287**, DOI 10.1017/S0143385711000988 - Béla Bollobás and Oliver Riordan,
*Sparse graphs: metrics and random models*, Random Structures Algorithms**39**(2011), no. 1, 1–38. MR**2839983**, DOI 10.1002/rsa.20334 - A. Carderi,
*Ultraproducts, weak equivalence and sofic entropy*, preprint, arxiv:1509.03189. - A. Carderi, D. Gaboriau, and M. de la Salle,
*Non-standard limits of graphs and some orbit equivalence invariants*, arxiv:1812.00704. - Gábor Elek,
*The combinatorial cost*, Enseign. Math. (2)**53**(2007), no. 3-4, 225–235. MR**2455943** - Damien Gaboriau,
*Coût des relations d’équivalence et des groupes*, Invent. Math.**139**(2000), no. 1, 41–98 (French, with English summary). MR**1728876**, DOI 10.1007/s002229900019 - Hamed Hatami, László Lovász, and Balázs Szegedy,
*Limits of locally-globally convergent graph sequences*, Geom. Funct. Anal.**24**(2014), no. 1, 269–296. MR**3177383**, DOI 10.1007/s00039-014-0258-7 - Alexander S. Kechris,
*Global aspects of ergodic group actions*, Mathematical Surveys and Monographs, vol. 160, American Mathematical Society, Providence, RI, 2010. MR**2583950**, DOI 10.1090/surv/160 - Marc Lackenby,
*Expanders, rank and graphs of groups*, Israel J. Math.**146**(2005), 357–370. MR**2151608**, DOI 10.1007/BF02773541 - Donald S. Ornstein and Benjamin Weiss,
*Ergodic theory of amenable group actions. I. The Rohlin lemma*, Bull. Amer. Math. Soc. (N.S.)**2**(1980), no. 1, 161–164. MR**551753**, DOI 10.1090/S0273-0979-1980-14702-3 - Klaus Schmidt,
*Amenability, Kazhdan’s property $T$, strong ergodicity and invariant means for ergodic group-actions*, Ergodic Theory Dynam. Systems**1**(1981), no. 2, 223–236. MR**661821**, DOI 10.1017/s014338570000924x

## 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.
- © Copyright 2019 American Mathematical Society
- 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
- MathSciNet review: 4069220