On Limit Eigenvalue Distributions Associated to Residual Chains of Groups

Let $G$ be a residually finite group. We give an explicit example in the discrete Heisenberg group that the Brown measure of multiplication operators $A\in \mathbb{Z}[G]\subseteq \mathcal{B}({\mathrm{\ell }}^2(G))$ in general can not be approximated using finite quotients $G\mathrm{/}N$ of $G$. We show that in finitely generated abelian groups the Brown measure can be approximated using finite quotients.