In what follows "homology" will mean group homology, i.e. $H_*(BG^\delta;{\mathbf R})$ for the group $G$ with the discrete topology.

It is well-known how to compute the homology of abelian groups, for example for a torsion-free abelian group A one has $H_n(A)=A\wedge_{\mathbf Z}A\wedge_{\mathbf Z}\ldots\wedge_{\mathbf Z}A$, with n factors.

Can one compute the homology of solvable groups, for example of the group $$\left\{\left(\begin{array}{cc}a&b\\ 0&a^{-1}\end{array}\right)\colon a,b\in{\mathbf C}\right\}$$ or of the group $$\left\{\left(\begin{array}{cc}a&b\\ 0&a^{-1}\end{array}\right)\colon a,b\in{\mathbf k}\right\}?$$ where $k\subset{\mathbf C}$ is some number field?

Milnor's Comm.Math.Helv.'83-paper gives an isomorphism $H_*(BG^\delta;{\mathbf Z}/p{\mathbf Z})=H_*(BG;{\mathbf Z}/p{\mathbf Z})$, but it's not clear to me what to do with this.

In what follows "homology" will mean group homology, i.e. $H_*(BG^\delta;{\mathbf R})$ for the group $G$ with the discrete topology.

It is well-known how to compute the homology of abelian groups, for example for a torsion-free abelian group A one has $H_n(A)=A\wedge_{\mathbf Z}A\wedge_{\mathbf Z}\ldots\wedge_{\mathbf Z}A$, with n factors.

Can one compute the homology of solvable groups, for example of the group $$\left\{\left(\begin{array}{cc}a&b\\ 0&a^{-1}\end{array}\right)\colon a,b\in{\mathbf C}\right\}$$ or of the group $$\left\{\left(\begin{array}{cc}a&b\\ 0&a^{-1}\end{array}\right)\colon a,b\in{\mathbf k}\right\}$$ where $k\subset{\mathbf C}$ is some number field?

If $G$ is a Lie group, Milnor's Comm.Math.Helv.'83-paper gives an isomorphism $H_*(BG^\delta;{\mathbf Z}/p{\mathbf Z})=H_*(BG;{\mathbf Z}/p{\mathbf Z})$, but it's not clear to me what to do with this and what might be the analogue for the number field case.