# gr.group theory – Sequence of epimorphisms of residually finite groups stabilizes

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”. 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 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?