On the growth of Betti numbers of locally symmetric spaces

By Miklós Abért, Nicolas Bergeron, Ian Biringer, Tsachik Gelander, Nikolay Nikolov, Jean Raimbault, Iddo Samet. 2011

58 citations recorded by the source corpus.

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 L\"uck Approximation Theorem \cite{luck}, which is much stronger than the linear upper bounds on Betti numbers given by Gromov in \cite{BGS}. The basic idea is to adapt the theory of local convergence, originally introduced for sequences of graphs of bounded degree by Benjamimi 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.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ea449110d41dc7d26d8c212fee5e8760c18961cb

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Families of L -Functions and Their Symmetry (2014, background): “10A general uniform such limit multiplicity theorem has been derived recently in [1].”
On the growth of $L^2$-invariants for sequences of lattices in Lie groups (2012, background): “As explained in the announcement [3], our original proof of Corollary 6.”
Random walks on random coset spaces with applications to Furstenberg entropy (2010, background): “This term was coined in [2].”