Strong limit multiplicity for arithmetic hyperbolic surfaces and 3-manifolds
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Strong limit multiplicity for arithmetic hyperbolic surfaces and 3-manifolds Mikołaj Fra˛czyk1
Received: 20 October 2017 / Accepted: 8 November 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We show that every sequence of torsion-free arithmetic congruence lattices in PGL(2, R) or PGL(2, C) satisfies a strong quantitative version of the limit multiplicity property. We deduce that for R > 0 in certain range, growing linearly in the degree of the invariant trace field, the volume of the R-thin part of any congruence arithmetic hyperbolic surface or congruence arithmetic hyperbolic 3-manifold M is of order at most Vol(M)11/12 . As an application we prove Gelander’s conjecture on homotopy type of arithmetic hyperbolic 3-manifolds: we show that there are constants A, B such that every such manifold M is homotopy equivalent to a simplicial complex with at most AVol(M) vertices, all of degrees bounded by B. Mathematics Subject Classification 30F40 · 20H10 · 22E46 · 11K16 Contents 1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Limit multiplicity property . . . . . . . . . . . . . 1.2 Benjamini–Schramm convergence . . . . . . . . . 1.3 Triangulations of arithmetic hyperbolic 3-manifolds
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This work was supported by a public grant as part of the Investissement d’avenir project, reference ANR-11-LABX-0056-LMH, LabEx mathématique Hadamard and by ERC Consolidator Grant No. 648017.
B Mikołaj Fra˛czyk
[email protected]; [email protected]
1
Alfréd Rényi Institute of Mathematics, Reáltanoda utca 13-15, Budapest 1053, Hungary
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M. Fra˛czyk 1.4 Growth of Betti numbers . . . . . . . . . . . . . . . . . . . . 1.5 Comparison with previous work . . . . . . . . . . . . . . . . 1.6 Outline of the paper . . . . . . . . . . . . . . . . . . . . . . . 1.7 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Quaternion algebras . . . . . . . . . . . . . . . . . . . . . . . 2.2 Congruence lattices . . . . . . . . . . . . . . . . . . . . . . . 2.3 Volume conventions . . . . . . . . . . . . . . . . . . . . . . . 3 Bilu equidistribution and consequences . . . . . . . . . . . . . . . 4 Trace formula for congruence lattices . . . . . . . . . . . . . . . . 4.1 Class groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Trace formula . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Non-Archimedean estimates . . . . . . . . . . . . . . . . . . . . . 5.1 Proposition 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Local orbital integrals . . . . . . . . . . . . . . . . . . . . . . 5.3 Character bounds . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Quasi-randomness and an estimate of Oγ (1V ) . . . . . . . . . 5.5 Representation zeta functions . . . . . . . . . . . . . . . . . . 5.6 Abelianization of U . . . . . . . . .
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