The mod 2 cohomology rings of congruence subgroups in the Bianchi groups
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The mod 2 cohomology rings of congruence subgroups in the Bianchi groups Ethan Berkove1
· Grant S. Lakeland2 · Alexander D. Rahm3
Received: 7 August 2018 / Accepted: 8 October 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract We establish a dimension formula involving a number of parameters for the mod 2 cohomology of finite index subgroups in the Bianchi groups (SL2 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
With an appendix by Bui Anh Tuan and Sebastian Schönnenbeck.
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Ethan Berkove [email protected] http://math.lafayette.edu/people/ethan-berkove/ Grant S. Lakeland [email protected] http://www.ux1.eiu.edu/~gslakeland/ Alexander D. Rahm [email protected] http://gaati.org/rahm/
1
Department of Mathematics, Lafayette College, 230 Pardee Hall, Easton, PA 18042, USA
2
Department of Mathematics and Computer Science, Eastern Illinois University, 600 Lincoln Avenue, Charleston, IL 61920, USA
3
Laboratoire de mathématiques GAATI, Université de la Polynésie Française, BP 6570, 98702 Faaa, French Polynesia
123
Journal of Algebraic Combinatorics
1 Introduction Calegari and Venkatesh have recently proven a numerical form of a Jacquet–Langlands correspondence for torsion classes on arithmetic hyperbolic 3-manifolds [6]. This can be seen as a new kind of Langlands programme, for which one has to study torsion in the cohomology of arithmetic groups. A class of arithmetic groups that is of natural interest here, as well as in the classical Langlands programme, consists of the congruence subgroups in the Bianchi groups. By a Bianchi group, we mean an SL2 group over the ring of integers in an imaginary quadratic number field. Our aim in this paper is to provide new tools for computing the torsion in the cohomology of the congruence subgroups in the Bianchi groups. There are already several approaches known for studying congruence subgroups and their cohomology: • Grunewald’s method of taking a presentation for the whole Bianchi group, and deriving presentations for finite index subgroups via the Reidemeister-Schreier algorithm [8]; • Utilizing the Eckmann–Shapiro lemma for computing cohomology of congruence subgroups directly from cohomological data of the full Bianchi group [20]; • Construction of a Voronoï cell complex for the congruence subgroup [4,21]. What one typically harvests with these approaches are tables of machine results in which everything looks somewhat ad hoc. A question posed by Fritz Grunewald to the third author asks for deeper structur
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