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.
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Alessandro Carderi 1
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@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},
}
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/
[1] Approximate equivalence of group actions, Ergodic Theory Dyn. Syst., Volume 38 (2018) no. 4, pp. 1201-1237 | DOI | MR | Zbl
[2] Parallelizability of proper actions, global
[3] A universal proper
[4] Convergence of normalized Betti numbers in nonpositive curvature (2018) (https://arxiv.org/abs/1811.02520v1)
[5] On the growth of
[6] 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] 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] Uniform rank gradient, cost and local-global convergence, Trans. Am. Math. Soc., Volume 373 (2020) no. 4, pp. 2311-2329 | DOI | MR | Zbl
[9] Manifolds of nonpositive curvature, Progress in Mathematics, 61, Birkhäuser, 1985 | DOI | Zbl
[10] Asymptotique des nombres de Betti, invariants
[11] Ultraproducts, weak equivalence and sofic entropy (2015) (https://arxiv.org/abs/1509.03189)
[12] On Farber sequences in locally compact groups (2018) (https://arxiv.org/abs/1812.05010)
[13] Non-standard limits of graphs and some orbit equivalence invariants, Ann. Henri Lebesgue, Volume 4 (2021), pp. 1235-1293 | DOI | MR | Zbl
[14] Orbit full groups for locally compact groups, Trans. Am. Math. Soc., Volume 370 (2018) no. 4, pp. 2321-2349 | DOI | MR | Zbl
[15] Riemannian geometry, Mathematics: Theory & Applications, Birkhäuser, 1992 (translated from the second Portuguese edition by Francis Flaherty) | MR | Zbl
[16] Hyperlinearity, essentially free actions and
[17] Geometry of growth: approximation theorems for
[18] On the virtual groups defined by ergodic actions of
[19] Measure theory. Vol. 1-2, 4, Torres Fremlin, Colchester, 2001-2003
[20] Coût des relations d’équivalence et des groupes, Invent. Math., Volume 139 (2000) no. 1, pp. 41-98 | DOI | Zbl
[21] Invariants
[22] Invariant descriptive set theory, Pure and Applied Mathematics (Boca Raton), 293, CRC Press, 2009 | Zbl
[23] Homotopy type and volume of locally symmetric manifolds, Duke Math. J., Volume 124 (2004) no. 3, pp. 459-515 | MR | Zbl
[24] Invariant random subgroups over non-Archimedean local fields, Math. Ann., Volume 372 (2018) no. 3-4, pp. 1503-1544 | DOI | MR | Zbl
[25] The structure of locally compact groups and Hilbert’s fifth problem, Trans. Am. Math. Soc., Volume 15 (1960), pp. 55-93 | MR | Zbl
[26] Algebraic topology, Cambridge University Press, 2002 | Zbl
[27]
[28] On Benjamini–Schramm limits of congruence subgroups, Isr. J. Math., Volume 239 (2020) no. 1, pp. 59-73 | DOI | MR | Zbl
[29] On the cost of generating an equivalence relation, Ergodic Theory Dyn. Syst., Volume 15 (1995) no. 6, pp. 1173-1181 | DOI | MR | Zbl
[30] Approximating
[31] 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] Approximating
[33] 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] Point realizations of transformation groups, Ill. J. Math., Volume 6 (1962), pp. 327-335 | MR | Zbl
[35]
[36]
[37] Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series, 35, Princeton University Press, 1997 | DOI | Zbl
[38] Ultraproducts of crossed product von Neumann algebras, Ill. J. Math., Volume 61 (2017) no. 3-4, pp. 275-286 | MR | Zbl
[39] Ergodic theory and semisimple groups, Monographs in Mathematics, 81, Birkhäuser, 1984 | DOI | Zbl
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