Equivariant K-homology of Bianchi groups with non-trivial class group

TL;DR

This paper computes the equivariant K-homology of Bianchi groups with non-trivial class groups, extending previous results limited to trivial class numbers, using algebraic and topological techniques involving $C^*$-algebras.

Contribution

It introduces a method to handle non-proper group actions on CW-complexes for Bianchi groups with non-trivial class groups, expanding the scope of equivariant K-homology calculations.

Findings

Computed K-theory of group $C^*$-algebras for Bianchi groups with non-trivial class group.

Extended previous results from trivial to non-trivial class number cases.

Developed a new approach using $C^*$-algebraic techniques and spectral sequences.

Abstract

We compute the equivariant K-homology of the groups PSL_2 of imaginary quadratic integers with trivial and non-trivial class-group. This was done before only for cases of trivial class number. We rely on reduction theory in the form of the $\Gamma$-CW-complex defined by Fl\"oge. We show that the difficulty arising from the non-proper action of $\Gamma$ on this complex can be overcome by considering a natural short exact sequence of $C^\ast$-algebras associated to the universal cover of the Borel-Serre compactification of the locally symmetric space associated to $\Gamma$. We use rather elementary $C^\ast$-algebraic techniques including a slightly modified Atiyah-Hirzebruch spectral sequence as well as several 6-term sequences. This computes the K-theory of the reduced and full group $C^\ast$-algebras of the Bianchi groups.