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

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

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.

Keywords

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

Mathematics Subject Classification

11F75 

Notes

Acknowledgements

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.

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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

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