Skip to main content
SpringerLink
Log in

Strong limit multiplicity for arithmetic hyperbolic surfaces and 3-manifolds

  • Published:
Inventiones mathematicae Aims and scope

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. There is a gap in the proof in [50], recently discovered by Nelson and Venkatesh [40, p. 103 footnote 7]. Fortunately the problem does not appear for \(G=\mathrm{PGL}(2,{{{\mathbb {R}}}}),\mathrm{PGL}(2,{{{\mathbb {C}}}}).\)

  2. We used here the fact that \({\mathbf {G}}(k)\) is an adjoint group.

  3. 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.

  4. In the previous version of the manuscript we proved a bound \(\zeta _U^*(7)\le \zeta _k(2)\prod _{\mathfrak {p}\in S\cup \mathrm{Ram}\,_f A}(N(\mathfrak {p})+1)\).

  5. non-torsion just to avoid disconnected centralizers.

  6. The formula is stated there for central simple division algebras but the same statement holds for any reductive algebraic group.

  7. If \(\gamma \) is hyperbolic is it unique, otherwise all eigenvalues are of modulus 1.

  8. i.e. distinct from 1.

  9. https://terrytao.wordpress.com/2013/07/18/a-cheap-version-of-the-kabatjanskii--levenstein-bound-for-almost-orthogonal-vectors/

  10. 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.

  11. 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.

References

  1. Abert, M., Bergeron, N., Biringer, I., Gelander, T., Nikolov, N., Raimbault, J., Samet, I.: On the growth of \(L^2\)-invariants for sequences of lattices in Lie groups. Ann. Math. 185(3), 711–790 (2017)

    MathSciNet  MATH  Google Scholar 

  2. Arthur, J. (ed.), Harmonic analysis, the trace formula, and shimura varieties. In: Clay Mathematics Proceedings, vol. 4. American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA

  3. Barbasch, D., Moscovici, H.: \(L^{2}\)-index and the Selberg trace formula. J. Funct. Anal. 53(2), 151–201 (1983)

    MathSciNet  MATH  Google Scholar 

  4. Bardestani, M., Mallahi-Karai, K., Salmasian, H.: Minimal dimension of faithful representations for \(p\)-groups. J. Group Theory 19(4), 589–608 (2016)

    MathSciNet  MATH  Google Scholar 

  5. Benjamini, I., Schramm, O.: Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6(23), 13 (2001) (electronic)

  6. Bergeron, N., Clozel, L.: Spectre automorphe des variétés hyperboliques et applications topologiques, Société mathématique de France (2005)

  7. Bilu, Y.: Limit distribution of small points on algebraic tori. Duke Math. J. 89(3), 465–476 (1997)

    MathSciNet  MATH  Google Scholar 

  8. Borel, A.: Commensurability classes and volumes of hyperbolic 3-manifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8(1), 1–33 (1981)

    MathSciNet  MATH  Google Scholar 

  9. Borel, A., Wallach, N.: Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, Second, Mathematical Surveys and Monographs, vol. 67. American Mathematical Society, Providence (2000)

    MATH  Google Scholar 

  10. Bott, R., Tu, L.W.: Differential Forms in Algebraic Topology, Graduate Texts in Mathematics, vol. 82. Springer, New York (1982)

    MATH  Google Scholar 

  11. Carayol, H.: Représentations cuspidales du groupe linéaire. Ann. Sci. École Norm. Sup. (4) 17(2), 191–225 (1984)

    MathSciNet  MATH  Google Scholar 

  12. Chinburg, T., Friedman, E.: The smallest arithmetic hyperbolic three-orbifold. Invent. Math. 86(3), 507–527 (1986)

    MathSciNet  MATH  Google Scholar 

  13. Clozel, L.: On limit multiplicities of discrete series representations in spaces of automorphic forms. Invent. Math. 83(2), 265–284 (1986)

    MathSciNet  MATH  Google Scholar 

  14. de George, D.L., Wallach, N.R.: Limit formulas for multiplicities in \(L^{2}({\varGamma } \backslash G)\). Ann. Math. (2) 107(1), 133–150 (1978)

    MathSciNet  Google Scholar 

  15. DeGeorge, D.L., Wallach, N.R.: Limit formulas for multiplicities in \(L^{2}({\varGamma } \backslash G)\). II. The tempered spectrum. Ann. Math. (2) 109(3), 477–495 (1979)

    MathSciNet  MATH  Google Scholar 

  16. DeGeorge, D.L.: On a theorem of Osborne and Warner. Multiplicities in the cuspidal spectrum. J. Funct. Anal. 48(1), 81–94 (1982)

    MathSciNet  MATH  Google Scholar 

  17. Delorme, P.: Formules limites et formules asymptotiques pour les multiplicités dans \({L}^2(g/\gamma )\). Duke Math. J. 53(3), 691–731 (1986). (French)

    MathSciNet  MATH  Google Scholar 

  18. Dixmier, J.: \({C}^*\)-algebras, North-Holland Publishing Co., Amsterdam-New York-Oxford. Translated from the French by Francis Jellett; North-Holland Mathematical Library, Vol. 15

  19. Dobrowolski, E.: On a question of Lehmer and the number of irreducible factors of a polynomial. Acta Arith. 34(4), 391–401 (1979)

    MathSciNet  MATH  Google Scholar 

  20. Finis, T.: The limit multiplicity problem for congruence subgroups of arithmetic lattices and the trace formula. Kyoto Proc. 2013, 164–176 (1871)

    Google Scholar 

  21. Finis, T., Lapid, E.: An approximation principle for congruence subgroups. J. Eur. Math. Soc. (JEMS) 20(5), 1075–1138 (2018)

    MathSciNet  MATH  Google Scholar 

  22. Finis, T., Lapid, E.: An approximation principle for congruence subgroups II: application to the limit multiplicity problem. Math. Z. 289(3–4), 1357–1380 (2018)

    MathSciNet  MATH  Google Scholar 

  23. Finis, T., Lapid, E., Müller, W.: Limit multiplicities for principal congruence subgroups of \({\rm GL}(n)\) and \({\rm SL}(n)\). J. Inst. Math. Jussieu 14(3), 589–638 (2015)

    MathSciNet  MATH  Google Scholar 

  24. Friedman, E.: Analytic formulas for the regulator of a number field. Invent. Math. 98(3), 599–622 (1989)

    MathSciNet  MATH  Google Scholar 

  25. Frączyk, M., Raimbault, J.: Betti numbers of Shimura curves and arithmetic three-orbifolds. Algebra Number Theory 13(10), 2359–2382 (2019)

    MathSciNet  MATH  Google Scholar 

  26. Gelander, T.: Homotopy type and volume of locally symmetric manifolds. Duke Math. J. 124(3), 459–515 (2004)

    MathSciNet  MATH  Google Scholar 

  27. Häsä, J., Stasinski, A.: Representation growth of compact linear groups. Trans. Am. Math. Soc. 372(2), 925–980 (2019)

    MathSciNet  MATH  Google Scholar 

  28. Labesse, J.-P., Langlands, R.P.: \(L\)-indistinguishability for \({\rm SL}(2)\). Can. J. Math. 31(4), 726–785 (1979)

    MathSciNet  MATH  Google Scholar 

  29. Lang, S.: Algebraic Number Theory, 2nd edn. Graduate Texts in Mathematics, vol. 110. Springer, New York

  30. Lang, S.: \({\rm SL}_2({\bf R})\), Graduate Texts in Mathematics, vol. 105. Springer, New York (1985). Reprint of the 1975 edition

  31. Langlands, R.P.: Base Change for GL(2), Annals of Mathematics Studies, vol. 96. Princeton University Press, Princeton; University of Tokyo Press, Tokyo (1980)

  32. Larsen, M.: http://mlarsen.math.indiana.edu (2008)

  33. Levit, A.: On Benjamini–Schramm limits of congruence subgroups (2017). arXiv:1705.04200

  34. Linowitz, B., McReynolds, D.B., Pollack, P., Thompson, L.: Bounded gaps between primes and the length spectra of arithmetic hyperbolic 3-orbifolds. C. R. Math. Acad. Sci. Paris 355(11), 1121–1126 (2017)

    MathSciNet  MATH  Google Scholar 

  35. Linowitz, B., McReynolds, D.B., Pollack, P., Thompson, L.: Systoles of arithmetic hyperbolic surfaces and 3-manifolds. Math. Res. Lett. 24(5), 1497–1522 (2017)

    MathSciNet  MATH  Google Scholar 

  36. Linowitz, B., McReynolds, D.B., Pollack, P., Thompson, L.: Counting and effective rigidity in algebra and geometry. Invent. Math. 213(2), 697–758 (2018)

    MathSciNet  MATH  Google Scholar 

  37. Matsushima, Y., et al.: A formula for the betti numbers of compact locally symmetric Riemannian manifolds. J. Differ. Geom. 1(1–2), 99–109 (1967)

    MathSciNet  MATH  Google Scholar 

  38. Matz, J.: Limit multiplicities for \({\rm SL}_2({\cal{O}}_F)\) in \({\rm SL}_2({\mathbb{R}}^{r_1}\oplus {\mathbb{C}}^{r_2})\). Groups Geom. Dyn. 13(3), 841–881 (2019)

    MathSciNet  Google Scholar 

  39. Milne, J.S.: Class Field Theory, lecture notes available at www.math.lsa.umich.edu (1997)

  40. Nelson, P.D., Venkatesh, A.: The orbit method and analysis of automorphic forms. arXiv preprint arXiv:1805.07750 (2018)

  41. Odlyzko, A.M.: Lower bounds for discriminants of number fields. II. Tôhoku Math. J. 29(2), 209–216 (1977)

    MathSciNet  MATH  Google Scholar 

  42. Ono, T.: Arithmetic of algebraic tori. Ann. Math. (2) 74, 101–139 (1961)

    MathSciNet  MATH  Google Scholar 

  43. Ono, T.: On the Tamagawa number of algebraic tori. Ann. Math. (2) 78, 47–73 (1963)

    MathSciNet  MATH  Google Scholar 

  44. Platonov, V., Rapinchuk, A.: Algebraic groups and number theory. Pure and Applied Mathematics, vol. 139. Academic Press, Inc., Boston, MA. Translated from the 1991 Russian original by Rachel Rowen

  45. Raimbault, J.: On the convergence of arithmetic orbifolds. Ann. Inst. Fourier (Grenoble) 67(6), 2547–2596 (2017)

    MathSciNet  MATH  Google Scholar 

  46. Ratcliffe, J.: Foundations of Hyperbolic Manifolds, Graduate Texts in Mathematics. Springer, New York (2006)

    Google Scholar 

  47. Reid, A., MacLachlan, C.: The Arithmetic of Hyperbolic 3-Manifolds, Graduate Texts in Mathematics, vol. 219. Springer, New York (2003)

    MATH  Google Scholar 

  48. Rohlfs, J., Speh, B.: On limit multiplicities of representations with cohomology in the cuspidal spectrum. Duke Math. J. 55(1), 199–211 (1987)

    MathSciNet  MATH  Google Scholar 

  49. Sarnak, P.C.: A note on the spectrum of cusp forms for congruence subgroups. Preprint (1982)

  50. Sauvageot, F.: Principe de densité pour les groupes réductifs. Compositio Math. 108(2), 151–184 (1997) (French, with English and French summaries)

  51. Savin, G.: Limit multiplicities of cusp forms. Invent. Math. 95(1), 149–159 (1989)

    MathSciNet  MATH  Google Scholar 

  52. Serre, J.-P.: Une "formule de masse" pour les extensions totalement ramifiées de degré donné d’un corps local. C. R. Acad. Sci. Paris Sér. A-B 286(22), A1031–A1036 (1978)

    Google Scholar 

  53. Shalom, Y.: Rigidity, unitary representations of semisimple groups, and fundamental groups of manifolds with rank one transformation group. Ann. Math. (2) 152(1), 113–182 (2000)

    MathSciNet  MATH  Google Scholar 

  54. Shyr, J.-M.: Class number formula of algebraic tori with applications to relative class numbers of certain relative quadratic extensions of algebraic number fields. Ph.D. Thesis. Thesis (Ph.D.)—The Johns Hopkins University

  55. Silberger, A.J.: \({\rm PGL}_{2}\) over the \(p\)-adics: its representations, spherical functions, and Fourier analysis, Lecture Notes in Mathematics, vol. 166. Springer, Berlin (1970)

  56. Silberger, A.J.: Irreducible representations of a maximal compact subgroup of \({\rm PGL}_{2}\) over the \(p\)-adics. Math. Ann. 229(1), 1–12 (1977)

    MathSciNet  MATH  Google Scholar 

  57. Ullmo, E., Yafaev, A.: Nombre de classes des tores de multiplication complexe et bornes inférieures pour les orbites galoisiennes de points spéciaux. Bull. Soc. Math. France 143(1), 197–228 (2015)

    MathSciNet  MATH  Google Scholar 

  58. Zimmer, R.J.: Ergodic Theory and Semisimple Groups, vol. 81. Springer, Berlin (2013)

    MATH  Google Scholar 

  59. Zimmert, R.: Ideale kleiner Norm in Idealklassen und eine Regulatorabschätzung. Invent. Math. 62(3), 367–380 (1981)

    MathSciNet  MATH  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. Alfréd Rényi Institute of Mathematics, Reáltanoda utca 13-15, Budapest, 1053, Hungary

    Mikołaj Frączyk

Authors
  1. Mikołaj Frączyk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mikołaj Frączyk.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-020-01021-1

Mathematics Subject Classification

Search

Navigation