Non-standard limits of graphs and some orbit equivalence invariants
Annales Henri Lebesgue, Volume 4 (2021), pp. 1235-1293.

Metadata

Keywords Orbit equivalence, Asymptotic Properties of Graphs and Groups, Ultraproducts, Cost, L2 Betti Numbers, Soficity, rank gradient

Abstract

We consider probability measure preserving discrete groupoids, group actions and equivalence relations in the context of general probability spaces. We study for these objects the notions of cost, 2 -Betti numbers, β-invariant and some higher-dimensional variants. We also propose various convergence results about 2 -Betti numbers and rank gradient for sequences of actions, groupoids or equivalence relations under weak finiteness assumptions. In particular we connect the combinatorial cost with the cost of the ultralimit equivalence relations. Finally a relative version of Stuck–Zimmer property is also considered.


References

[ABB + 17] Abért, Miklos; Bergeron, Nicolas; Biringer, Ian; Gelander, Tsachik; Nikolov, Nikolay; Raimbault, Jean; Samet, Iddo On the growth of L 2 -invariants for sequences of lattices in Lie groups, Ann. Math., Volume 185 (2017) no. 3, pp. 711-790 | DOI | MR | Zbl

[ADR00] Anantharaman-Delaroche, Claire; Renault, Jean N. Amenable groupoids, Monographies de l’Enseignement Mathématique, 36, L’Enseignement Mathématique, Université de Genève, 2000 | MR | Zbl

[AFS19] Alekseev, Vadim; Finn-Sell, Martin Sofic boundaries of groups and coarse geometry of sofic approximations, Groups Geom. Dyn., Volume 13 (2019) no. 1, pp. 191-234 | DOI | MR | Zbl

[AGN17] Abért, Miklos; Gelander, Tsachik; Nikolov, Nikolay Rank, combinatorial cost, and homology torsion growth in higher rank lattices, Duke Math. J., Volume 166 (2017) no. 15, pp. 2925-2964 | MR | Zbl

[Alv08] Alvarez, Aurélien Une théorie de Bass–Serre pour les relations d’équivalence et les groupoïdes boréliens, Ph. D. Thesis, ENS-Lyon, France (2008) (2008ENSL0458)

[AN12] Abért, Miklos; Nikolov, Nikolay Rank gradient, cost of groups and the rank versus Heegaard genus problem, J. Eur. Math. Soc., Volume 14 (2012) no. 5, pp. 1657-1677 | MR | Zbl

[AP18] Aaserud, Andreas N.; Popa, Sorin Approximate equivalence of group actions, Ergodic Theory Dyn. Syst., Volume 38 (2018) no. 4, pp. 1201-1237 | DOI | MR | Zbl

[AT20] Abért, Miklos; Tóth, Lázló M. Uniform rank gradient, cost and local-global convergence, Trans. Am. Math. Soc., Volume 373 (2020) no. 4, pp. 2311-2329 | DOI | MR | Zbl

[AW13] Abért, Miklos; Weiss, Benjamin Bernoulli actions are weakly contained in any free action, Ergodic Theory Dyn. Syst., Volume 33 (2013) no. 2, pp. 323-333 | DOI | MR | Zbl

[BG04] Bergeron, Nicolas; Gaboriau, Damien Asymptotique des nombres de Betti, invariants l 2 et laminations, Comment. Math. Helv., Volume 79 (2004) no. 2, pp. 362-395 | DOI | MR | Zbl

[Bil86] Billingsley, Patrick P. Probability and measure, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, 1986 | Zbl

[Car11] Carderi, Alessandro Cost for measured groupoids, Ph. D. Thesis, Ecole normale supérieure de lyon - ENS LYON, France (2011)

[Car15] Carderi, Alessandro Ultraproducts, weak equivalence and sofic entropy (2015) (https://arxiv.org/abs/1509.03189)

[CKTD13] Conley, Clinton T.; Kechris, Alexander S.; Tucker-Drob, Robin D. Ultraproducts of measure preserving actions and graph combinatorics, Ergodic Theory Dyn. Syst., Volume 33 (2013) no. 2, pp. 334-374 | DOI | MR | Zbl

[Cor17] Cordeiro, Luiz An elementary approach to sofic groupoids (2017) (https://arxiv.org/abs//1708.08023)

[CP17] Creutz, Darren; Peterson, Jesse Stabilizers of ergodic actions of lattices and commensurators, Trans. Am. Math. Soc., Volume 369 (2017) no. 6, pp. 4119-4166 | DOI | MR | Zbl

[EL10] Elek, Gábor; Lippner, Gábor Sofic equivalence relations, J. Funct. Anal., Volume 258 (2010) no. 5, pp. 1692-1708 | DOI | MR | Zbl

[Ele07] Elek, Gábor The combinatorial cost, Enseign. Math., Volume 53 (2007) no. 3-4, pp. 225-235 | MR | Zbl

[Ele10a] Elek, Gábor Betti numbers are testable, Fete of combinatorics and computer science (Katona, Gyula O. H. et al., eds.) (Bolyai Society Mathematical Studies), Volume 20, Springer; János Bolyai Mathematical Society, Budapest, 2010, pp. 139-149 (Selected papers of the conference held in Keszthely, Hungary, August 11–15, 2008 dedicated to László Lovász on the occasion of his 60th birthday) | DOI | MR | Zbl

[Ele10b] Elek, Gábor Parameter testing in bounded degree graphs of subexponential growth, Random Struct. Algorithms, Volume 37 (2010) no. 2, pp. 248-270 | DOI | MR | Zbl

[ES05] Elek, Gábor; Szabó, Endre Hyperlinearity, essentially free actions and L 2 -invariants. The sofic property, Math. Ann., Volume 332 (2005) no. 2, pp. 421-441 | DOI | MR | Zbl

[Far98] Farber, Michael Geometry of growth: approximation theorems for L 2 invariants, Math. Ann., Volume 311 (1998) no. 2, pp. 335-375 | DOI | MR | Zbl

[FM77] Feldman, Jacob; Moore, Calvin C. Ergodic equivalence relations, cohomology, and von Neumann algebras. I, Trans. Am. Math. Soc., Volume 234 (1977) no. 2, pp. 289-324 | DOI | MR | Zbl

[Fre04] Fremlin, David H. Measure Theory: Measure algebras. Volume 3, Colchester, Torres Fremlin, 2004 (corrected second printing of the 2002 original) | Zbl

[Gab98] Gaboriau, Damien Mercuriale de groupes et de relations, C. R. Acad. Sci. Paris Sér. I Math., Volume 326 (1998) no. 2, pp. 219-222 | DOI | MR | Zbl

[Gab00] Gaboriau, Damien Coût des relations d’équivalence et des groupes, Invent. Math., Volume 139 (2000) no. 1, pp. 41-98 | DOI | MR | Zbl

[Gab02] Gaboriau, Damien Invariants l 2 de relations d’équivalence et de groupes, Publ. Math., Inst. Hautes Étud. Sci., Volume 95 (2002), pp. 93-150 | DOI | MR | Zbl

[Gab05a] Gaboriau, Damien Examples of groups that are measure equivalent to the free group, Ergodic Theory Dyn. Syst., Volume 25 (2005) no. 6, pp. 1809-1827 | DOI | MR | Zbl

[Gab05b] Gaboriau, Damien Invariant percolation and harmonic Dirichlet functions, Geom. Funct. Anal., Volume 15 (2005) no. 5, pp. 1004-1051 | DOI | MR | Zbl

[HLS14] Hatami, Hamed; Lovász, Lázló; Szegedy, Balázs Limits of locally-globally convergent graph sequences, Geom. Funct. Anal., Volume 24 (2014) no. 1, pp. 269-296 | DOI | MR | Zbl

[Kai19] Kaiser, Tom Combinatorial cost: a coarse setting, Trans. Am. Math. Soc., Volume 372 (2019) no. 4, pp. 2855-2874 | DOI | MR | Zbl

[Kec95] Kechris, Alexander S. Classical descriptive set theory, Graduate Texts in Mathematics, 156, Springer, 1995 | MR | Zbl

[Kec10] Kechris, Alexander S. Global aspects of ergodic group actions, Mathematical Surveys and Monographs, 160, American Mathematical Society, 2010 | MR | Zbl

[KM04] Kechris, Alexander S.; Miller, Benjamin D. Topics in orbit equivalence, Lecture Notes in Mathematics, 1852, Springer, 2004 | MR | Zbl

[Lev95] Levitt, Gilbert On the cost of generating an equivalence relation, Ergodic Theory Dyn. Syst., Volume 15 (1995) no. 6, pp. 1173-1181 | DOI | MR | Zbl

[LN35] Lusin, Nikolaĭ N.; Novikoff, Pëtr S. Choix effectif d’un point dans un complementaire analytique arbitraire, donné par un crible, Fundam. Math., Volume 25 (1935), pp. 559-560 | DOI | Zbl

[LO11] Lück, Wolfgang; Osin, Denis V. Approximating the first L 2 -Betti number of residually finite groups, J. Topol. Anal., Volume 3 (2011) no. 2, pp. 153-160 | DOI | MR | Zbl

[Lüc94] Lück, Wolfgang Approximating L 2 -invariants by their finite-dimensional analogues, Geom. Funct. Anal., Volume 4 (1994) no. 4, pp. 455-481 | DOI | MR | Zbl

[Neu32] von Neumann, John Einige Sätze über messbare Abbildungen, Ann. Math., Volume 33 (1932) no. 3, pp. 574-586 | DOI | Zbl

[Oza09] Ozawa, Narukata Hyperlinearity, sofic groups and applications to group theory, 2009 (Unpublished notes, https://www.kurims.kyoto-u.ac.jp/~narutaka/notes/NoteSofic.pdf)

[PSV20] Popa, Sorin; Shlyakhtenko, Dimitri L.; Vaes, Stefaan Classification of regular subalgebras of the hyperfinite II 1 factor, J. Math. Pures Appl., Volume 140 (2020), pp. 280-308 | DOI | Zbl

[Roe03] Roe, John Lectures on coarse geometry, University Lecture Series, 31, American Mathematical Society, 2003 | MR | Zbl

[Sau05] Sauer, Roman L 2 -Betti numbers of discrete measured groupoids, Int. J. Algebra Comput., Volume 15 (2005) no. 5-6, pp. 1169-1188 | DOI | MR | Zbl

[Sch20] Schrödl-Baumann, Michael 2 -Betti numbers of random rooted simplicial complexes, Manuscr. Math., Volume 162 (2020) no. 3-4, pp. 284-304 | MR | Zbl

[SZ94] Stuck, Garett; Zimmer, Robert J. Stabilizers for ergodic actions of higher rank semisimple groups, Ann. Math., Volume 139 (1994) no. 3, pp. 723-747 | DOI | MR | Zbl

[Tak02] Takesaki, Masamichi Theory of operator algebras. I, Encyclopaedia of Mathematical Sciences, 124, Springer, 2002 reprint of the first (1979) edition, also present in the Operator Algebras and Non-commutative Geometry series, Vol. 5 | Zbl

[Tak15] Takimoto, Atsushi L 2 -Betti numbers and costs in the framework of discrete groupoids (2015) (https://arxiv.org/abs/1502.01555)

[Tho08] Thom, Andreas Sofic groups and Diophantine approximation, Commun. Pure Appl. Math., Volume 61 (2008) no. 8, pp. 1155-1171 | DOI | MR | Zbl

[Ued06] Ueda, Yoshimichi Notes on treeability and costs for discrete groupoids in operator algebra framework, Operator Algebras: The Abel Symposium 2004. Proceedings of the first Abel symposium, Oslo, Norway, September 3–5, 2004 (Bratelli, Ola et al., eds.) (Abel Symposia), Volume 1, Springer, 2006, pp. 259-279 | DOI | MR | Zbl