Asymptotic invariants of lattices in locally compact groups

Alessandro Carderi

doi:10.5802/crmath.417

Geometry and Topology, Group theory

Asymptotic invariants of lattices in locally compact groups

Alessandro Carderi ¹

¹ A.C., Institut für Algebra und Geometrie, KIT, 76128 Karlsruhe, Germany

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 375-415.

Abstract

The aim of this work is to understand some of the asymptotic properties of sequences of lattices in a fixed locally compact group. In particular we will study the asymptotic growth of the Betti numbers of the lattices renormalized by the covolume and the rank gradient, the minimal number of generators also renormalized by the covolume. For doing so we will consider the ultraproduct of the sequence of actions of the locally compact group on the coset spaces and we will show how the properties of one of its cross sections are related to the asymptotic properties of the lattices.

Received: 2021-06-21
Revised: 2022-08-16
Accepted: 2022-09-01
Published online: 2023-01-12

DOI: 10.5802/crmath.417

Author's affiliations:

Alessandro Carderi ¹

¹ A.C., Institut für Algebra und Geometrie, KIT, 76128 Karlsruhe, Germany

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{CRMATH_2023__361_G1_375_0,
     author = {Alessandro Carderi},
     title = {Asymptotic invariants of lattices in locally compact groups},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {375--415},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.417},
     language = {en},
}

TY  - JOUR
AU  - Alessandro Carderi
TI  - Asymptotic invariants of lattices in locally compact groups
JO  - Comptes Rendus. Mathématique
PY  - 2023
SP  - 375
EP  - 415
VL  - 361
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.417
LA  - en
ID  - CRMATH_2023__361_G1_375_0
ER  -

%0 Journal Article
%A Alessandro Carderi
%T Asymptotic invariants of lattices in locally compact groups
%J Comptes Rendus. Mathématique
%D 2023
%P 375-415
%V 361
%I Académie des sciences, Paris
%R 10.5802/crmath.417
%G en
%F CRMATH_2023__361_G1_375_0

Alessandro Carderi. Asymptotic invariants of lattices in locally compact groups. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 375-415. doi : 10.5802/crmath.417. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.417/

References
Cited by

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

[2] Herbert Abels Parallelizability of proper actions, global $K$ -slices and maximal compact subgroups, Math. Ann., Volume 212 (1974), pp. 1-19 | DOI | MR | Zbl

[3] Herbert Abels A universal proper $G$ -space, Math. Z., Volume 159 (1978) no. 2, pp. 143-158 | DOI | MR | Zbl

[4] Miklós Abért; Nicolas Bergeron; Ian Biringer; Tsachik Gelander Convergence of normalized Betti numbers in nonpositive curvature (2018) (https://arxiv.org/abs/1811.02520v1)

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

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

[7] Miklós Abért; Nikolay Nikolov 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

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

[9] Werner Ballmann; Mikhael Gromov; Viktor Schroeder Manifolds of nonpositive curvature, Progress in Mathematics, 61, Birkhäuser, 1985 | DOI | Zbl

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

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

[12] Alessandro Carderi On Farber sequences in locally compact groups (2018) (https://arxiv.org/abs/1812.05010)

[13] Alessandro Carderi; Damien Gaboriau; Mickael de la Salle Non-standard limits of graphs and some orbit equivalence invariants, Ann. Henri Lebesgue, Volume 4 (2021), pp. 1235-1293 | DOI | MR | Zbl

[14] Alessandro Carderi; François Le Maître Orbit full groups for locally compact groups, Trans. Am. Math. Soc., Volume 370 (2018) no. 4, pp. 2321-2349 | DOI | MR | Zbl

[15] Manfredo Perdigão do Carmo Riemannian geometry, Mathematics: Theory & Applications, Birkhäuser, 1992 (translated from the second Portuguese edition by Francis Flaherty) | MR | Zbl

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

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

[18] Peter Forrest On the virtual groups defined by ergodic actions of $R^{n}$ and $Z^{n}$ , Adv. Math., Volume 14 (1974), pp. 271-308 | DOI | MR | Zbl

[19] David H. Fremlin Measure theory. Vol. 1-2, 4, Torres Fremlin, Colchester, 2001-2003

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

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

[22] Su Gao Invariant descriptive set theory, Pure and Applied Mathematics (Boca Raton), 293, CRC Press, 2009 | Zbl

[23] Tsachik Gelander Homotopy type and volume of locally symmetric manifolds, Duke Math. J., Volume 124 (2004) no. 3, pp. 459-515 | MR | Zbl

[24] Tsachik Gelander; Arie Levit Invariant random subgroups over non-Archimedean local fields, Math. Ann., Volume 372 (2018) no. 3-4, pp. 1503-1544 | DOI | MR | Zbl

[25] Viktor M. Gluškov The structure of locally compact groups and Hilbert’s fifth problem, Trans. Am. Math. Soc., Volume 15 (1960), pp. 55-93 | MR | Zbl

[26] Allen Hatcher Algebraic topology, Cambridge University Press, 2002 | Zbl

[27] David Kyed; Henrik Densing Petersen; Stefaan Vaes $L^{2}$ -Betti numbers of locally compact groups and their cross section equivalence relations, Trans. Am. Math. Soc., Volume 367 (2015) no. 7, pp. 4917-4956 | DOI | MR | Zbl

[28] Arie Levit On Benjamini–Schramm limits of congruence subgroups, Isr. J. Math., Volume 239 (2020) no. 1, pp. 59-73 | DOI | MR | Zbl

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

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

[31] Wolfgang Lück Survey on classifying spaces for families of subgroups, Infinite groups: geometric, combinatorial and dynamical aspects (Progress in Mathematics), Volume 248, Birkhäuser, 2005, pp. 269-322 | MR | Zbl

[32] Wolfgang Lück Approximating $L^{2}$ -invariants by their classical counterparts, EMS Surv. Math. Sci., Volume 3 (2016) no. 2, pp. 269-344 | DOI | MR | Zbl

[33] Wolfgang Lück; David Meintrup On the universal space for group actions with compact isotropy, Geometry and topology: Aarhus (1998) (Contemporary Mathematics), Volume 258, American Mathematical Society, 2000, pp. 293-305 | MR | Zbl

[34] George W. Mackey Point realizations of transformation groups, Ill. J. Math., Volume 6 (1962), pp. 327-335 | MR | Zbl

[35] Henrik Densing Petersen $L^{2}$ -Betti numbers of locally compact groups, C. R. Math. Acad. Sci. Paris, Volume 351 (2013) no. 9-10, pp. 339-342 | DOI | MR | Zbl

[36] Henrik Densing Petersen; Roman Sauer; Andreas Thom $L^{2}$ -Betti numbers of totally disconnected groups and their approximation by Betti numbers of lattices, J. Topol., Volume 11 (2018) no. 1, pp. 257-282 | DOI | MR | Zbl

[37] William P. Thurston Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series, 35, Princeton University Press, 1997 | DOI | Zbl

[38] Reiji Tomatsu Ultraproducts of crossed product von Neumann algebras, Ill. J. Math., Volume 61 (2017) no. 3-4, pp. 275-286 | MR | Zbl

[39] Robert J. Zimmer Ergodic theory and semisimple groups, Monographs in Mathematics, 81, Birkhäuser, 1984 | DOI | Zbl

Cited by Sources:

Comments - Policy