Group Theory/Geometry
On the growth of Betti numbers of locally symmetric spaces
[Comportement des nombres de Betti des espaces localement symétriques]
Comptes Rendus. Mathématique, Tome 349 (2011) no. 15-16, pp. 831-835.

Nous annonçons de nouveaux résultats concernant le comportement asymptotique des nombres de Betti des espaces localement symétriques de rang supérieur lorsque leurs volumes tendent vers lʼinfini. Notre résultat principal – une version uniforme du théorème dʼapproximation de Lück (1994) [10] – est plus fort que la majoration linéaire en le volume obtenue par Gromov dans Ballmann et al. (1985) [3].

Lʼidée de base est dʼadapter la théorie de la convergence locale, initialement introduite pour les suites de graphes de degré borné par Benjamimi et Schramm, à des suites de variétés riemanniennes. Lʼutilisation de théorèmes de rigidité nous permet de montrer que lorsque le volume tend vers lʼinfini, les variétés convergent localement vers le revêtement universel de manière assez forte pour en déduire la convergence des nombres de Betti normalisés par le volume.

We announce new results concerning the asymptotic behavior of the Betti numbers of higher rank locally symmetric spaces as their volumes tend to infinity. Our main theorem is a uniform version of the Lück Approximation Theorem (Lück, 1994 [10]) which is much stronger than the linear upper bounds on Betti numbers given by Gromov in Ballmann et al. (1985) [3].

The basic idea is to adapt the theory of local convergence, originally introduced for sequences of graphs of bounded degree by Benjamini and Schramm, to sequences of Riemannian manifolds. Using rigidity theory we are able to show that when the volume tends to infinity, the manifolds locally converge to the universal cover in a sufficiently strong manner that allows us to derive the convergence of the normalized Betti numbers.

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DOI : 10.1016/j.crma.2011.07.013
Abert, Miklos 1 ; Bergeron, Nicolas 2 ; Biringer, Ian 3 ; Gelander, Tsachik 4 ; Nikolov, Nikolay 5 ; Raimbault, Jean 2 ; Samet, Iddo 4

1 Alfréd Rényi Institute of Mathematics, POB 127, H-1364 Budapest, Hungary
2 Institut de mathématiques de Jussieu, unité mixte de recherche 7586 du CNRS, université Pierre-et-Marie-Curie, 4, place Jussieu, 75252 Paris cedex 05, France
3 Yale University, Mathematics Department, PO Box 208283, New Haven, CT 06520, USA
4 Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
5 Department of Mathematics, Imperial College London, SW7 2AZ, UK
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     title = {On the growth of {Betti} numbers of locally symmetric spaces},
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Abert, Miklos; Bergeron, Nicolas; Biringer, Ian; Gelander, Tsachik; Nikolov, Nikolay; Raimbault, Jean; Samet, Iddo. On the growth of Betti numbers of locally symmetric spaces. Comptes Rendus. Mathématique, Tome 349 (2011) no. 15-16, pp. 831-835. doi : 10.1016/j.crma.2011.07.013. http://www.numdam.org/articles/10.1016/j.crma.2011.07.013/

[1] Miklos Abert, Nicolas Bergeron, Ian Biringer, Tsachik Gelander, Nikolay Nikolov, Jean Raimbault, Iddo Samet, in preparation.

[2] Miklos Abert, Yair Glasner, Balint Virag, The measurable Kesten theorem, Preprint.

[3] Ballmann, Werner; Gromov, Mikhael; Schroeder, Viktor Manifolds of Nonpositive Curvature, Progress in Mathematics, vol. 61, Birkhäuser Boston Inc., Boston, MA, 1985

[4] Barbasch, Dan; Moscovici, Henri L2-index and the Selberg trace formula, J. Funct. Anal., Volume 53 (1983) no. 2, pp. 151-201

[5] Benjamini, Itai; Schramm, Oded Recurrence of distributional limits of finite planar graphs, Electron. J. Probab., Volume 23 (2001) no. 6, p. 13 (electronic)

[6] Bergeron, Nicolas; Venkatesh, Akshay The asymptotic growth of torsion homology for arithmetic groups | arXiv

[7] Borel, Armand Density properties for certain subgroups of semi-simple groups without compact components, Ann. of Math. (2), Volume 72 (1960), pp. 179-188

[8] Chabauty, Claude Limite dʼensembles et géométrie des nombres, Bull. Soc. Math. France, Volume 78 (1950), pp. 143-151

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

[10] Lück, W. Approximating L2-invariants by their finite-dimensional analogues, Geom. Funct. Anal., Volume 4 (1994) no. 4, pp. 455-481

[11] Margulis, G.A. Discrete Subgroups of Semisimple Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Results in Mathematics and Related Areas (3), vol. 17, Springer-Verlag, Berlin, 1991

[12] Hee, Oh Uniform pointwise bounds for matrix coefficients of unitary representations and applications to Kazhdan constants, Duke Math. J., Volume 113 (2002) no. 1, pp. 133-192

[13] Stuck, Garrett; Zimmer, Robert J. Stabilizers for ergodic actions of higher rank semisimple groups, Ann. of Math. (2), Volume 139 (1994) no. 3, pp. 723-747

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