Congruence Link Complements—A 3-Dimensional Rademacher Conjecture

  • M. D. Baker
  • A. W. ReidEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 245)


In this article we discuss a 3-dimensional version of a conjecture of Rademacher concerning genus 0 congruence subgroups of \(\mathrm{PSL}(2,\mathbb Z)\). We survey known results, as well as including some new results that make partial progress on the conjecture.


Link complement Bianchi group Congruence subgroup 



The work contained in this paper was developed over multiple visits to the University of Texas by the first author, and the Université de Rennes 1 by the second author. We also wish to thank the Université Paul Sabatier, the Max Planck Institüt, Bonn, I.C.T.P. Trieste and The Institute for Advanced Study for their support and hospitality as this work unfolded over several years. We also wish to thank several people without whose help we would not have been able to complete this work: M. D. E. Conder, D. Holt, M. Goerner, E. O’Brien, A. Page, M. H. Sengun and A. Williams. We also thank I. Agol, N. Hoffman and P. Sarnak for useful conversations on topics related to this work. We are also very grateful to the referee for their comments and helpful suggestions.


  1. 1.
    Adams, C.C., Reid, A.W.: Systoles of hyperbolic \(3\)-manifolds. Math. Proc. Camb. Philos. Soc. 128, 103–110 (2000)Google Scholar
  2. 2.
    Baker, M.D.: Link complements and quadratic imaginary number fields, Ph.D. thesis M.I.T. (1981)Google Scholar
  3. 3.
    Baker, M.D.: Link complements and the homology of arithmetic subgroups of PSL \((2,{C})\), I.H.E.S. preprint (1982)Google Scholar
  4. 4.
    Baker, M.D.: Link complements and integer rings of class number greater than one. In: Topology ’90, pp. 55–59. Ohio State University, Mathematical Research Institute Publications 1, de Gruyter (1992)Google Scholar
  5. 5.
    Baker, M.D.: Link complements and the Bianchi modular groups. Trans. Am. Math. Soc. 353, 3229–3246 (2001)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Baker, M.D.: Commensurability classes of arithmetic link complements. J. Knot Theory Ramif. 10, 943–957 (2001)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Baker, M.D., Reid, A.W.: Principal congruence link complements. Ann. Fac. Sci. Toulouse 23, 1063–1092 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Baker, M.D., Reid, A.W.: Principal congruence links: Class number greater than \(1\). Exp. Math. (2017). Scholar
  9. 9.
    Baker, M.D., Goerner, M., Reid, A.W.: All principal congruence link complements, in preparationGoogle Scholar
  10. 10.
    Bergeron, N.: Torsion homology growth in arithmetic groups, to appear in the proceedings of the 7th European Congress of MathematicsGoogle Scholar
  11. 11.
    Bergeron, N., Venkatesh, A.: The asymptotic growth of torsion homology for arithmetic groups. J. Inst. Math. Jussieu 12, 391–447 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Bergeron, N., Sengun, M.H., Venkatesh, A.: Torsion homology growth and cycle complexity of arithmetic manifolds. Duke Math. J. 165, 1629–1693 (2016)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Blume-Nienhaus, J.: Lefschetzzahlen fur Galois-Operationen auf der Kohomologie arithmetischer Gruppen, Ph.D. thesis, Universität Bonn, Bonn Math. Publications, 230 (1991)Google Scholar
  14. 14.
    Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24, 235–265 (1997)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Chua, K.S., Lang, M.L., Yang, Y.: On Rademachers conjecture: congruence subgroups of genus zero of the modular group. J. Algebr. 277, 408–428 (2004)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Dennin, J.B.: Fields of modular functions of genus \(0\). Ill. J. Math. 15, 442–455 (1971)Google Scholar
  17. 17.
    Dennin, J.B.: Subfields of \(K(2^{n})\) of genus \(0\). Ill. J. Math. 16, 502–518 (1972)Google Scholar
  18. 18.
    Dennin, J.B.: The genus of subfields of \(K(p^{n})\). Ill. J. Math. 18, 246–264 (1974)Google Scholar
  19. 19.
    Dixon, L.E.: Linear Groups, with an Exposition of the Galois Field Theory. Dover Publications, New York (1958)Google Scholar
  20. 20.
    Futer, D., Kalfagianni, E., Purcell, J.: On diagrammatic bounds of knot volumes and spectral invariants. Geom. Dedicata 147, 115–130 (2010)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Goerner, M.: Visualizing regular tessellations: principal congruence links and equivariant morphisms from surfaces to \(3\)-manifold, Ph.D. thesis, U. C. Berkeley (2011)Google Scholar
  22. 22.
    Goerner, M.: Regular tessellation link complements. Exp. Math. 24, 225–246 (2015)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Gordon, C.McA: Links and their complements. Topology and Geometry: Commemorating SISTAG. Contemporary Mathematics, vol. 314, pp. 71–82. American Mathematical Society, Providence (2002)Google Scholar
  24. 24.
    Grunewald, F., Schwermer, J.: Arithmetic quotients of hyperbolic \(3\)- space, cusp forms and link complements. Duke Math. J. 48, 351–358 (1981)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Grunewald, F., Schwermer, J.: A non-vanishing theorem for the cuspidal cohomology of \({\rm SL}_{2}\) over imaginary quadratic integers. Math. Ann. 258, 183–200 (1981)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Hatcher, A.: Hyperbolic structures of arithmetic type on some link complements. J. Lond. Math. Soc. 27, 345–355 (1983)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Lackenby, M.: Spectral geometry, link complements and surgery diagrams. Geom. Dedicata 147, 191–206 (2010)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Lakeland, G., Leininger, C.: Systoles and Dehn surgery for hyperbolic \(3\)- manifolds. Algebr. Geomet. Topol. 14, 1441–1460 (2014)Google Scholar
  29. 29.
    Le, T.: Growth of homology torsion in finite coverings and hyperbolic volume, to appear Annales de l’Institut FourierGoogle Scholar
  30. 30.
    Maclachlan, C., Reid, A.W.: The Arithmetic of Hyperbolic \(3\)-Manifolds. Graduate Texts in Mathematics, vol. 219. Springer, Berlin (2003)Google Scholar
  31. 31.
    Martelli, B., Petronio, C.: Dehn filling of the “Magic" \(3\)-manifold. Commun. Anal. Geom. 14, 969–1026 (2006)Google Scholar
  32. 32.
    Page, A.: Computing arithmetic Kleinian groups. Math. Comput. 84, 2361–2390 (2015)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Palaparthi, S.: Closed geodesic lengths in hyperbolic link complements in \(S^3\). Int. J. Pure Appl. Math. 83, 45–53 (2013)Google Scholar
  34. 34.
    Rohlfs, J.: On the cuspidal cohomology of the Bianchi modular groups. Math. Z. 188, 253–269 (1985)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Sebbar, A.: Classification of torsion-free genus zero congruence subgroups. Proc. Am. Math. Soc. 129, 2517–2527 (2001)CrossRefGoogle Scholar
  36. 36.
    Sengun, M.H.: On the integral cohomology of Bianchi groups. Exp. Math. 20, 487–505 (2011)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Stephan, J.: Complémentaires d’entrelacs dans \(S^3\) et ordres maximaux des algèbres de quaternions. C. R. Acad. Sci. Paris Sér. I Math. 322, 685–688 (1996)Google Scholar
  38. 38.
    Stephan, J.: On arithmetic hyperbolic links. J. Knot Theory Ramif. 8, 373–389 (1999)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Swan, R.G.: Generators and relations for certain special linear groups. Adv. Math. 6, 1–77 (1971)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Thompson, J.G.: A finiteness theorem for subgroups of \({\rm PSL} (2,{\mathbb{R}})\). Proc. Symp. Pure Math. 37, 533–555 (1980). A.M.S. PublicationsGoogle Scholar
  41. 41.
    Thurston, W.P.: The Geometry and Topology of \(3\)-Manifolds, Princeton University mimeographed notes (1979)Google Scholar
  42. 42.
    Thurston, W.P.: Three-dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Am. Math. Soc. 6, 357–381 (1982)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Vogtmann, K.: Rational homology of Bianchi groups. Math. Ann. 272, 399–419 (1985)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Zograf, P.: A spectral proof of Rademacher’s conjecture for congruence subgroups of the modular group. J. Reine Angew. Math. 414, 113–116 (1991)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.IRMARUniversité de Rennes 1Rennes CedexFrance
  2. 2.Department of MathematicsRice UniversityHoustonUSA

Personalised recommendations