On the growth of Betti numbers of locally symmetric spaces

Abert, Miklos; Bergeron, Nicolas; Biringer, Ian; Gelander, Tsachik; Nikolov, Nikolay; Raimbault, Jean; Samet, Iddo

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.

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

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