On the equivariant K-homology of PSL\_2 of the imaginary quadratic integers

TL;DR

This paper computes the torsion part of the equivariant K-homology for Bianchi groups using a new representation ring splitting technique, linking algebraic topology with number theory.

Contribution

Introduces a novel representation ring splitting method for computing Bredon homology, applied to Bianchi groups' equivariant K-homology.

Findings

Derived explicit formulas for torsion in equivariant K-homology of Bianchi groups.

Connected algebraic topology computations with elementary number-theoretic quantities.

Extended the torsion subcomplex reduction technique to Bredon homology.

Abstract

We establish formulae for the part due to torsion of the equivariant K-homology of all the Bianchi groups (PSL\_2 of the imaginary quadratic integers), in terms of elementary number-theoretic quantities. To achieve this, we introduce a novel technique in the computation of Bredon homology: representation ring splitting, which allows us to adapt the recent technique of torsion subcomplex reduction from group homology to Bredon homology.