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I'm looking for an example of a finitely presented and finitely generated amenable group, that has a subgroup which is not finitely generated.

The question is easy for finitely generated amenable group and an example is the lamp-lighter group $C_2\wr \mathbb{Z}$.

An abelian Abelian and finitely generated group has no such subgroups. There exists a bigger class of groups with this property?

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# Example of an amenable finitely generated and presented group with a non-finitely generated subgroup

I'm looking for an example of a finitely presented and finitely generated amenable group, that has a subgroup which is not finitely generated.

The question is easy for finitely generated amenable group and an example is the lamp-lighter group $C_2\wr \mathbb{Z}$.

An abelian and finitely generated group has no such subgroups. There exists a bigger class of groups with this property?