Abstract
We continue our overview of the various notions of geometric and spectral convergence with the notion of convergence in the sense of Benjamini and Schramm.
Originally introduced for sequences of discrete graphs, whose size tends to infinity while the degrees of the vertices remain bounded, the notion was adapted to the case of sequences of metric spaces by the " seven samurai " (Abert-Bergeron-Biringer-Gelander-Nikolov-Raimbault-Samet) and by Lewis Bowen. It has mainly been exploited in the case of sequences of locally symmetric spaces. We shall see that convergence in the sense of Benjamini and Schramm implies convergence of empirical spectral measures as well as convergence of renormalized Betti numbers.
References
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