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On the convergence of arithmetic orbifolds

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arxiv 1311.5375 v2 pith:X4MEAZ64 submitted 2013-11-21 math.GT math.GRmath.NT

Jean Raimbault This is my paper · ORCID
classification math.GTmath.GRmath.NT
keywords hyperbolicarithmeticorbifoldsproductsequencesapplicationarbitrarybenjamini--schramm
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We discuss the geometry of some arithmetic orbifolds locally isometric to a product of real hyperbolic spaces of dimension two and three, and prove that certain sequences of non-uniform orbifolds are convergent to this space in a geometric ("Benjamini--Schramm") sense for hyperbolic three--space and a product of hyperbolic planes. We also deal with arbitrary 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.

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    For non-lattice discrete subgroups of higher-rank simple Lie groups, the maximal injectivity radius on balls of radius r grows at least c log log log log r.

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