Abstract
We show that every sequence of torsion-free arithmetic congruence lattices in \(\mathrm{PGL}(2,{{{\mathbb {R}}}})\) or \(\mathrm{PGL}(2,{{{\mathbb {C}}}})\) 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 \(\mathrm{Vol}(M)^{11/12}\). 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 \(A\mathrm{Vol}(M)\) vertices, all of degrees bounded by B.
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Notes
We used here the fact that \({\mathbf {G}}(k)\) is an adjoint group.
The statement is actually true all regular semisimple elements, but in the torsion case the centralizer might be bigger than what we wrote in the proof.
In the previous version of the manuscript we proved a bound \(\zeta _U^*(7)\le \zeta _k(2)\prod _{\mathfrak {p}\in S\cup \mathrm{Ram}\,_f A}(N(\mathfrak {p})+1)\).
non-torsion just to avoid disconnected centralizers.
The formula is stated there for central simple division algebras but the same statement holds for any reductive algebraic group.
If \(\gamma \) is hyperbolic is it unique, otherwise all eigenvalues are of modulus 1.
i.e. distinct from 1.
If \({{{\mathbb {K}}}}={{{\mathbb {R}}}}\) they are defined over \({{{\mathbb {Q}}}}\) and if \({{{\mathbb {K}}}}={{{\mathbb {C}}}}\) the are defined over a quadratic imaginary number field.
The key feature used for non-uniform lattices is that they are all defined over a quadratic imaginary field. This implies a uniform lower bound on the lengths of closed geodesics on the quotients.
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Acknowledgements
This began as a part of author’s PhD thesis at the Université Paris-Sud. I would like to thank my supervisor Emmanuel Breuillard for suggesting this problem as well as for many useful remarks. I am grateful to Nicolas Bergeron and Erez Lapid for careful reading the first version of the manuscript. I acknowledge the support of ERC Consolidator Grant No. 648017 during the last stages of work. I am thankful to the Institute for Advanced Study for providing excellent working conditions when I wrote the current version of the manuscpript. Finally I thank anonymous referees whose valuable remarks and suggestions led to a much improved exposition and improvement of results.
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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.
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Frączyk, M. Strong limit multiplicity for arithmetic hyperbolic surfaces and 3-manifolds. Invent. math. 224, 917–985 (2021). https://doi.org/10.1007/s00222-020-01021-1
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DOI: https://doi.org/10.1007/s00222-020-01021-1