As is described in the title, is there a known example such that there is a surjective homomorphism of groups $$f: G\rightarrow H,$$ with $G$ and $H$ finitely presented, $G$ is residually finite, and $H$ is non-residually finite, such that $\ker f$ is finitely generated?

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    There are very concrete examples of matrix groups $G$ with this property. See for example the paper: Yves de Cornulier, Finitely presentable, non-Hopfian groups with Kazhdan’s property (T) and infinite outer automorphism group, Proc. Amer. Math. Soc. 135 (2007), no. 4, 951-959. – Andreas Thom Oct 22 at 11:27
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    (1) Abels constructed in 1978 a finitely presented, residually finite (linear), solvable group $G$ with a cyclic central subgroup $Z$ such that $G/Z$ is not residually finite. (2) The paper of mine quoted by Andreas is a similar construction, with "solvable" replaced with "with Kazhdan's Property T". (3) Ben's answer gives examples answering the question with $G$ hyperbolic (in which case we can't expect the kernel to be cyclic or central). – YCor Oct 22 at 11:48
  • @AndreasThom Thank you for your comments. – Bruno Oct 22 at 12:01
  • @YCor Thank you so much for your explanation! – Bruno Oct 22 at 12:02
up vote 10 down vote accepted

You can obtain such a surjection for any finitely presented non-residually finite group $H$ using Daniel Wise's residually finite Rips construction, which is the main result of this paper: A Residually Finite Version of Rips's Construction.

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