On the convergence of arithmetic orbifolds
Annales de l'Institut Fourier, Volume 67 (2017) no. 6, p. 2547-2596
We discuss the geometry of some arithmetic orbifolds locally isometric to a product X of real hyperbolic spaces m of dimension m=2,3, and prove that certain sequences of non-compact orbifolds are convergent to X in a geometric (“Benjamini–Schramm”) sense for low-dimensional cases (when X is equal to 2 × 2 or 3 ). We also deal with sequences of maximal arithmetic three–dimensional hyperbolic lattices defined over a quadratic or cubic field. A motivating application is the study of Betti numbers of Bianchi groups.
Cet article est consacré à l’étude de la géométrie globale de certaines orbi-variétés localement isométriques à un produit d’espaces tridimensionnels et de plans hyperboliques. On démontre que pour les petites dimensions (pour l’espace ou le plan hyperbolique, ou un produit de plans hyperboliques) certaines suites de telles orbi-variétés non-compactes de volume fini convergent vers l’espace symétrique en un sens géométrique précis (« convergence de Benjamini–Schramm »). On traite aussi le cas des réseaux arithmétiques maximaux en dimension trois dont les corps de traces sont quadratiques ou cubiques. Une des principales motivations est d’étudier l’asymptotique des nombres de Betti des groupes de Bianchi.
Received : 2014-01-10
Revised : 2014-09-17
Accepted : 2017-03-08
Published online : 2017-12-14
DOI : https://doi.org/10.5802/aif.3143
Classification:  22E40,  11F75,  11F72,  57M27
Keywords: Arithmetic hyperbolic manifolds, Limit multiplicities, Three–dimensional manifolds
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     author = {Raimbault, Jean},
     title = {On the convergence of arithmetic orbifolds},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {67},
     number = {6},
     year = {2017},
     pages = {2547-2596},
     doi = {10.5802/aif.3143},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2017__67_6_2547_0}
}
Raimbault, Jean. On the convergence of arithmetic orbifolds. Annales de l'Institut Fourier, Volume 67 (2017) no. 6, pp. 2547-2596. doi : 10.5802/aif.3143. https://aif.centre-mersenne.org/item/AIF_2017__67_6_2547_0/

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