The homological torsion of PSL_2 of the imaginary quadratic integers

TL;DR

This paper establishes a new connection between the homological torsion of Bianchi groups and geometric invariants, enabling explicit computations of their group homology and K-theory via a novel torsion subcomplex reduction technique.

Contribution

It introduces a new method, torsion subcomplex reduction, to compute the homological torsion and geometric invariants of Bianchi groups, and applies it to determine their K-theory.

Findings

Explicit computation of the integral group homology of Bianchi groups.

Establishment of a correspondence between homological torsion and geometric invariants.

Determination of the K-theory of reduced C*-algebras of Bianchi groups.

Abstract

Denote by Q(sqrt{-m}), with m a square-free positive integer, an imaginary quadratic number field, and by A its ring of integers. The Bianchi groups are the groups SL_2(A). We reveal a correspondence between the homological torsion of the Bianchi groups and new geometric invariants, which are effectively computable thanks to their action on hyperbolic space. We expose a novel technique, the torsion subcomplex reduction, to obtain these invariants. We use it to explicitly compute the integral group homology of the Bianchi groups. Furthermore, this correspondence facilitates the computation of the equivariant K-homology of the Bianchi groups. By the Baum/Connes conjecture, which is verified by the Bianchi groups, we obtain the K-theory of their reduced C*-algebras in terms of isomorphic images of their equivariant K-homology.