Uniform rank gradient, cost, and local-global convergence

Abért, Miklós; Tóth, László

doi:10.1090/tran/8008

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