# Residually finite group surjective to nonresidually finite group with finitely generated kernel

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?

• 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 '18 at 11:27
• (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 '18 at 11:48
• @AndreasThom Thank you for your comments. – Bruno Oct 22 '18 at 12:01
• @YCor Thank you so much for your explanation! – Bruno Oct 22 '18 at 12:02

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.