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Residually finite group surjective to nonresidually finite group with finitely generated kernel
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As is described in the title, is there a known example such that there is a surjective homomorphism of groups $$f: Grightarrow H,$$ with $G$ and $H$ finitely presented, $G$ is residually finite, and $H$ is non-residually finite, such that $ker f$ is finitely generated?
gr.group-theory geometric-group-theory
edited 39 mins ago
Martin Sleziak
2,78432028
asked 2 hours ago
Bruno
686
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up vote
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As is described in the title, is there a known example such that there is a surjective homomorphism of groups $$f: Grightarrow H,$$ with $G$ and $H$ finitely presented, $G$ is residually finite, and $H$ is non-residually finite, such that $ker f$ is finitely generated?
gr.group-theory geometric-group-theory
edited 39 mins ago
Martin Sleziak
2,78432028
asked 2 hours ago
Bruno
686
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
32 mins ago
(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
12 mins ago
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
As is described in the title, is there a known example such that there is a surjective homomorphism of groups $$f: Grightarrow H,$$ with $G$ and $H$ finitely presented, $G$ is residually finite, and $H$ is non-residually finite, such that $ker f$ is finitely generated?
gr.group-theory geometric-group-theory
edited 39 mins ago
Martin Sleziak
2,78432028
asked 2 hours ago
Bruno
686
As is described in the title, is there a known example such that there is a surjective homomorphism of groups $$f: Grightarrow H,$$ with $G$ and $H$ finitely presented, $G$ is residually finite, and $H$ is non-residually finite, such that $ker f$ is finitely generated?
gr.group-theory geometric-group-theory
gr.group-theory geometric-group-theory
edited 39 mins ago
Martin Sleziak
2,78432028
asked 2 hours ago
Bruno
686
edited 39 mins ago
Martin Sleziak
2,78432028
asked 2 hours ago
Bruno
686
edited 39 mins ago
Martin Sleziak
2,78432028
edited 39 mins ago
Martin Sleziak
2,78432028
edited 39 mins ago
Martin Sleziak
2,78432028
2,78432028
asked 2 hours ago
Bruno
686
asked 2 hours ago
Bruno
686
asked 2 hours ago
Bruno
686
686
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
32 mins ago
(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
12 mins ago
add a comment |Â
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
32 mins ago
(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
12 mins ago
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
32 mins ago
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
32 mins ago
(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
12 mins ago
(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
12 mins ago
add a comment |Â
1 Answer
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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.
answered 47 mins ago
Ben Barrett
213
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
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.
answered 47 mins ago
Ben Barrett
213
New contributor
Ben Barrett is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |Â
up vote
2
down vote
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.
answered 47 mins ago
Ben Barrett
213
New contributor
Ben Barrett is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |Â
up vote
2
down vote
up vote
2
down vote
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.
answered 47 mins ago
Ben Barrett
213
New contributor
Ben Barrett is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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.
answered 47 mins ago
Ben Barrett
213
New contributor
Ben Barrett is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 47 mins ago
Ben Barrett
213
New contributor
Ben Barrett is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 47 mins ago
Ben Barrett
213
answered 47 mins ago
Ben Barrett
213
213
New contributor
Ben Barrett is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Ben Barrett is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Ben Barrett is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |Â
add a comment |Â
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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
32 mins ago
(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
12 mins ago