The mod 2 cohomology rings of congruence subgroups in the Bianchi groups

  • Ethan BerkoveEmail author
  • Grant S. Lakeland
  • Alexander D. Rahm


We establish a dimension formula involving a number of parameters for the mod 2 cohomology of finite index subgroups in the Bianchi groups (SL\(_2\) groups over the ring of integers in an imaginary quadratic number field). The proof of our formula involves an analysis of the equivariant spectral sequence, combined with torsion subcomplex reduction. We also provide an algorithm to compute a Ford domain for congruence subgroups in the Bianchi groups from which the parameters in our formula can be explicitly computed.


Cohomology of arithmetic groups Fundamental domains Congruence subgroup Bianchi group Special linear group over imaginary quadratic integers 

Mathematics Subject Classification




The authors are grateful to the “Groups in Galway” conference series; the plans for this paper were made during a meeting which was co-organized by the third author, and attended by the second author and the Appendix’s second author. The second author thanks Alan Reid for introducing him to this topic and for numerous helpful conversations. The third is thankful for being supported by Gabor Wiese’s University of Luxembourg grant AMFOR. Special thanks go to Norbert Krämer for helpful suggestions on our manuscript. We would like to thank an anonymous referee, whose useful comments improved the quality of this paper.


  1. 1.
    Adem, A., Milgram, R.J.: Cohomology of Finite Groups, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 309, 2nd edn. Springer, Berlin (2004)Google Scholar
  2. 2.
    Berkove, E., Rahm, A.D.: The mod 2 cohomology rings of \(\text{ SL }_2\) of the imaginary quadratic integers. J. Pure Appl. Algebra 220(3), 944–975 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bui, A.T., Rahm, A.D., Wendt, M.: The Farrell–Tate and Bredon homology for \({\rm PSL}_{4}({\mathbb{Z}})\) via cell subdivisions. J. Pure Appl. Algebra 223(7), 2872–2888 (2019)Google Scholar
  4. 4.
    Braun, O., Coulangeon, R., Nebe, G., Schönnenbeck, S.: Computing in arithmetic groups with Voronoï’s algorithm. J. Algebra 435, 263–285 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Brown, K.S.: Cohomology of Groups, Graduate Texts in Mathematics, vol. 87. Springer, New York (1994). Corrected reprint of the 1982 originalGoogle Scholar
  6. 6.
    Calegari, F., Venkatesh, A.: A torsion Jacquet–Langlands correspondence. Astérisque (409), x+226 (2019)Google Scholar
  7. 7.
    Ellis, G.: Homological Algebra Programming, Computational Group Theory and the Theory of Groups, Contemporary Mathematics, vol. 470, pp. 63–74. American Mathematical Society, Providence (2008)CrossRefGoogle Scholar
  8. 8.
    Grunewald, F., Schwermer, J.: Subgroups of Bianchi groups and arithmetic quotients of hyperbolic 3-space. Trans. Amer. Math. Soc. 335(1), 47–78 (1993)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Henn, H.-W.: The cohomology of SL(3, Z[1/2]). K-Theory 16(4), 299–359 (1999)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Klein, F.: Ueber binäre Formen mit linearen Transformationen in sich selbst. Math. Ann. 9(2), 183–208 (1875)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Krämer, N.: Imaginärquadratische Einbettung von Ordnungen rationaler Quaternionenalgebren, und die nichtzyklischen endlichen Untergruppen der Bianchi-Gruppen. Preprint (German). (2017)
  12. 12.
    Maskit, B.: Kleinian Groups, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 287. Springer, Berlin (1988)Google Scholar
  13. 13.
    Moerdijk, I., Svensson, J.-A.: The equivariant Serre spectral sequence. Proc. Amer. Math. Soc. 118(1), 263–278 (1993)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Lascurain Orive, A. The shape of the Ford domains for \(\Gamma _0\) (N). Conform. Geom. Dyn. 3, 1–23 (1999). (electronic)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Page, A.: Computing arithmetic Kleinian groups. Math. Comp. 84(295), 2361–2390 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Rahm, A.D.: The homological torsion of \(\text{ PSL }_2\) of the imaginary quadratic integers. Trans. Amer. Math. Soc. 365(3), 1603–1635 (2013)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Rahm, A.D.: Homology and K-theory of the Bianchi groups. C. R. Math. Acad. Sci. Paris 349(11–12), 615–619 (2011)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Rahm, A.D.: Accessing the cohomology of discrete groups above their virtual cohomological dimension. J. Algebra 404, 152–175 (2014)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Rahm, A.D., Fuchs, M.: The integral homology of \(\text{ PSL }_2\) of imaginary quadratic integers with nontrivial class group. J. Pure Appl. Algebra 215(6), 1443–1472 (2011)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Rahm, A.D., Tsaknias, P.: Genuine Bianchi modular forms of higher level, at varying weight and discriminant. J. Théor. Nombres Bordeaux 31(1), 27–48 (2019)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Schönnenbeck, S.: Resolutions for unit groups of orders. J. Homotopy Relat. Struct. 12(4), 837–852 (2017)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Schwermer, J., Vogtmann, K.: The integral homology of \(\text{ SL }_2\) and \(\text{ PSL }_2\) of Euclidean imaginary quadratic integers. Comment. Math. Helv. 58(4), 573–598 (1983)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Singer, W.M.: Steenrod Squares in Spectral Sequences, Mathematical Surveys and Mongraphs, vol. 129. American Mathematical Society (AMS), Providence (2006)CrossRefGoogle Scholar
  24. 24.
    Soulé, C.: The cohomology of \(\text{ SL }_{3}\)(Z). Topology 17(1), 1–22 (1978)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Vogtmann, K.: Rational homology of Bianchi groups. Math. Ann. 272(3), 399–419 (1985)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsLafayette CollegeEastonUSA
  2. 2.Department of Mathematics and Computer ScienceEastern Illinois UniversityCharlestonUSA
  3. 3.Laboratoire de mathématiques GAATIUniversité de la Polynésie FrançaiseFaaaFrench Polynesia

Personalised recommendations