It's, by now, more or less well known that residual finiteness is not a quasi-isometry invariant for finitely generated groups (see here for an example). Thus the following question makes sense:
Question: given a finitely generated group $G$, is there a finitely generated group $H$ such that $H$ is residually finite and $G$ and $H$ are quasi-isometric? That is, does the class of groups quasi-isometric to $G$ always contain residually finite groups?
Acknowledging this question is fairly open-ended, I would be glad with any reference in this direction. Thanks in advance. The following tries to motivate the above via some C*-algebraic questions.
Motivation (and guess): Should the above be known true, then one would be able to manually build so-called quasi-diagonalizing projections for the reduced group C*-algebra of $G$, that is, finite rank projections $p_n \in \mathcal{B}(\ell^2 G)$, with $p_n \rightarrow 1$ in the strong operator topology and $$ ||p_n \lambda_g - \lambda_g p_n|| \rightarrow 0 $$ for every $g \in G$, where $\lambda$ is the left regular representation of $G$. Indeed, note that if $H$ is QI to $G$ and res. finite and amenable, then, by work of Orfanos there are projections $q_n$ for $H$ as above, and those can be pullback-ed to projections $p_n$ for $G$. Thus, since this C*-question is still open, my guess is that the original question is open as well.
