It's, by now, more or less well known that residual finiteness is not a quasi-isometry invariant for finitely generated groups (see [here](https://mathoverflow.net/questions/324465/is-residual-finiteness-a-quasi-isometry-invariant-for-f-g-groups) 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.




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**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.



Existence of a quasi-isometric residually finite group?
Take any finitely-presented group $G$ with undecidable word problem. Then $G$ is not quasi-isometric to any finitely generated group with decidable word problem, in particular, to any residually-finite group. (Note that finite presentability and decidability of the WP are quasi-isometry invariant. The latter is because quasi-isometries preserve the equivalence class of the Dehn function and WP is for a finitely-presented group decidable iff the Dehn function is recursive.) 


