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The mod 2 cohomology rings of congruence subgroups in the Bianchi groups. (English) Zbl 1472.11164

The authors establish a formula for the dimension of the mod \(2\) cohomology for a Bianchi group \(\Gamma\). Besides certain invariants of \(\Gamma\) itself, this formula needs as input also the mod \(2\) Betti numbers of the locally symmetric space associated to \(\Gamma\). The authors’ approach uses non-central torsion subcomplex reduction as in previous work of the first- and third-named authors, [J. Pure Appl. Algebra 220, No. 3, 944–975 (2016; Zbl 1401.11100)], together with an algorithm for constructing a fundamental domain of \(\Gamma\) inside hyperbolic \(3\)-space. This algorithm differs from A. Page’s [Math. Comput. 84, No. 295, 2361–2390 (2015; Zbl 1330.11026)], in that it also gives a cell structure in which every cell stabilizer fixes its cell pointwise, a datum that is needed for the authors’ purposes.

MSC:

11F75 Cohomology of arithmetic groups
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