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

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.

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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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With an appendix by Bui Anh Tuan and Sebastian Schönnenbeck.

Appendix A: Machine computations

Appendix A: Machine computations

We use Sebastian Schönnenbeck’s implementation [21] of the Voronoï cell complex to compute the cohomology of a sample (six levels \(\eta \) in each of ten imaginary quadratic fields) of congruence subgroups \(\Gamma _0(\eta )\). Then, we use Bui Anh Tuan’s implementation of Rigid Facets Subdivision in order to extract the non-central 2-torsion subcomplex. This allows us to check our example computations with the algorithm of Sect. 6, and to illustrate which values the parameters in our formulas can take.

Table 1 Cases \(\Gamma _0(\eta )\) in the sample with non-empty non-central 2-torsion subcomplex
Table 2 Cases \(\Gamma _0(\eta )\) in the sample with empty non-central 2-torsion subcomplex

Let \(\mathcal {O}_{-m}\) be the ring of integers in the field \({\mathbb {Q}}(\sqrt{-m})\) with discriminant \(\Delta \). We present the ideal \(\eta \subset \mathcal {O}_{-m}\) with the smallest possible number of generators; hence, when we use two generators, it is because \(\eta \) is not principal. We let c be the co-rank defined in Sect. 2.3; let \(\beta _q = \dim _{{\mathbb {Q}}}H_q(_{\Gamma _0(\eta )} \backslash \mathcal {H}; \,{\mathbb {Q}})\) and \(\beta ^q = \dim _{{\mathbb {F}}_2}H^q(_{\Gamma _0(\eta )} \backslash \mathcal {H}; \,{\mathbb {F}}_2)\) for \(q = 1, 2\); and \(_{\Gamma _0(\eta )} \backslash X_s\) the orbit space of the non-central 2-torsion subcomplex. We further write \(H^q := \dim _{{\mathbb {F}}_2}H^q({\Gamma _0(\eta )} ; \,{\mathbb {F}}_2)\). Let r be the rank of the \(d_2^{0,1}\)-differential of the equivariant spectral sequence. In many cases, notably for \(c = 0\), or when there are no components different from in \(_{\Gamma _0(\eta )} \backslash X_s\), the \(d_2^{0,2+4k}\)-differential vanishes because of the lemmata in Sect. 5. Note that for one case in our sample, \(\Gamma _0(\langle 2, \sqrt{-6}\rangle )\) in SL\(_2({\mathbb {Z}}[\sqrt{-6}\,])\), the machine computation yields rank\((d_2^{0, 2+4k}) = 1\). In all other cases in our sample, the computation yields \(d_2^{0, 2+4k} = 0\), so we do not print \(d_2^{0, 2+4k}\). The machine calculations in HAP [7] allow us to produce \(H^q\) and \(\beta _1\) directly from Sebastian Schönnenbeck’s cell complexes. From \(H^q\) and \(\beta _1\), we use the corollaries in Sect. 5 to get \(\beta ^1\), r and c in Table 1. When \(_{\Gamma _0(\eta )} \backslash X_s\) is empty, then because of Proposition 18, only the Betti numbers are of interest; results for those cases can be found in Table 2. In all cases except for the Eisenstein integers in \({\mathbb {Q}}(\sqrt{-3}\,)\), the Euler characteristic \(\chi \) of \(_{\Gamma _0(\eta )} \backslash \mathcal {H}\) vanishes [25], so we only need \(\beta ^1\). Therefore, we indicate \(\chi = 1-\beta _1+\beta _2\) only in those Eisenstein integers cases, where it can be nonzero.

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Berkove, E., Lakeland, G.S. & Rahm, A.D. The mod 2 cohomology rings of congruence subgroups in the Bianchi groups. J Algebr Comb 52, 527–560 (2020). https://doi.org/10.1007/s10801-019-00912-8

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Keywords

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

Mathematics Subject Classification

  • 11F75