Let $G\_1 \to G\_2 \to \cdots$ be a sequence of epimorphisms of finitely generated residually finite groups. Does it eventually stabilize? That is, are all but finitely many epimorphisms actually isomorphisms?


Note that finitely generated residually finite groups are Hopfian, so this excludes the simple counterexample of each $G\_i$ being a fixed group and each epimorphisms being a fixed one onto itself.


The analogous result holds when the groups are residually free: this is Proposition 6.8 in Charpentier Guirardel ["Limit groups as limits of free groups"](https://eclass.uoa.gr/modules/document/file.php/MATH154/%CE%88%CE%B3%CE%B3%CF%81%CE%B1%CF%86%CE%B1%20%CE%A0%CE%B1%CE%BB%CE%B1%CE%B9%CF%8E%CE%BD%20%CE%A3%CE%B5%CE%BC%CE%B9%CE%BD%CE%B1%CF%81%CE%AF%CF%89%CE%BD/Guirardel-Champetier.pdf). The proof only uses the fact that residually free groups are residually $SL\_2(\mathbb{C})$, and it seems that it can be adapted to the case where each $G\_i$ is residually $GL\_n(\mathbb{C})$ for a fixed $n$. It seems unlikely that this holds for general residaully finite groups: the [Jordan-Schur Theorem](https://groupprops.subwiki.org/wiki/Jordan-Schur_theorem_on_abelian_normal_subgroups_of_small_index) implies that for a general finite group the minimal degree $n$ such that it embeds into $GL\_n(\mathbb{C})$ can be arbitrarily large.


Is there another way to adapt the proof? Is there a counterexample?



Sequence of epimorphisms of residually finite groups stabilizes
The answer is "no". The lamplighter group (which is infinitely presented) is a limit of a sequence of virtually free groups and surjective homomorphisms (see, for example, this [question and answers there](https://mathoverflow.net/questions/75784/is-there-a-non-hopfian-lacunary-hyperbolic-group)). All virtually free groups are residually finite.


