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Can you give me an example of a finitely generated infinitely presented amenable group which is a quotient of a finitely presented amenable group?

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    This is just an idea...but the group given by $\langle a, b, c; a^2, [b, c], [ab, a], abac^{-1}a^{-1}\rangle$ is metabelian and contains the lamplighter group (which is infinitely presented). This is a result of Baumslag, but I can't find a decent reference. So...you could see what the normal closure of the lamplighter group in this is, and see if it will give you an infinite presentation...(i.e. is it infinitely generated as a normal subgroup).2011-12-05
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    (Or you could just try and take a finite subset of the relators for the lamplighter group and see what happens - this seems easier, but all my attempts have spectacularly failed!)2011-12-05
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    Lamplighter will not work because of a Theorem of R.Bieri and R.strebel which says: If $G$ is an infinitely presented metabelian group then any finitely presented covering group contains a nonabelian free group.2011-12-05

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An answer was given by Mark Sapir here:

https://mathoverflow.net/questions/82819/quotients-of-f-p-amenable-groups

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    You could accept your own answer so that this thread is considered as answered by the engine... :)2012-01-05