NUMBER-THEORETIC FORMULAE FOR THE COHOMOLOGY RING STRUCTURE OF THE BIANCHI GROUPS

Alexander D. Rahm (1)

1 NUIG - Department of Mathematics

We describe the torsion in the homology of Bianchi groups, i.e. PSL2 (o) for o the ring of integers in an imaginary quadratic number field. We show that the 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 o. 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.

Domaines: K-théorie et homologie [math.KT]

Alexander D. Rahm. NUMBER-THEORETIC FORMULAE FOR THE COHOMOLOGY RING STRUCTURE OF THE BIANCHI GROUPS. 2011. ⟨hal-00618167v3⟩