Equivariant K-homology and K-theory for some discrete planar affine groups

TL;DR

This paper computes the equivariant K-homology and K-theory for specific discrete affine groups, verifying the Baum-Connes conjecture through explicit calculations involving models for classifying spaces and torsion analysis.

Contribution

It provides explicit computations of both sides of the Baum-Connes conjecture for certain affine groups, including models for classifying spaces and torsion subgroup analysis.

Findings

Explicit generators for K_0 are identified and matched by the assembly map.

A 3-dimensional model for the classifying space is constructed, facilitating calculations.

The conjecture is verified for the considered groups through detailed K-theoretic analysis.

Abstract

We consider the semi-direct products $G=\mathbb Z^2\rtimes GL_2(\mathbb Z), \mathbb Z^2\rtimes SL_2(\mathbb Z)$ and $\mathbb Z^2\rtimes\Gamma(2)$ (where $\Gamma(2)$ is the congruence subgroup of level 2). For each of them, we compute both sides of the Baum-Connes conjecture, namely the equivariant $K$-homology of the classifying space $\underline{E}G$ for proper actions on the left-hand side, and the analytical K-theory of the reduced group $C^*$-algebra on the right-hand side. The computation of the LHS is made possible by the existence of a 3-dimensional model for $\underline{E}G$, which allows to replace equivariant K-homology by Bredon homology. We pay due attention to the presence of torsion in $G$, leading to an extensive study of the wallpaper groups associated with finite subgroups. For the second and third groups, the computations in $K_0$ provide explicit generators that are matched by the Baum-Connes assembly map.