Number-theoretic formulae for the cohomology ring structure of the Bianchi groups

Details Number-theoretic formulae for the cohomology ring structure of the Bianchi groups Number-theoretic formulae for the cohomology ring structure of the Bianchi groups

2011-12-19

arxiv.org/abs/1112.4262v1

Mathematics

K-Theory and Homology

Scientific paper

The Bianchi groups are the arithmetic groups PSL_2 over rings of integers in imaginary quadratic number fields. We show that their homological torsion is completely determined by the numbers of conjugacy classes of finite subgroups of the Bianchi groups, by detaching information from geometric models and expressing it only with the group structure. Formulae for the numbers of conjugacy classes of finite subgroups have been determined in a thesis of Kraemer, in terms of elementary number-theoretic information on the ring of integers. An evaluation of these formulae for a large number of Bianchi groups is provided numerically. Our new insights about the homological torsion allow us to give a conceptual description of the cohomology ring structure of the Bianchi groups.

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