It is known that quotients of finitely generated groups are finitely generated and that the quotient of a finitely presented group is finitely presented iff the normal subgroup is the normal closure of a finite subset of the group. Hence: $$G~\text{is }F_1\Rightarrow G/N~\text{is }F_1 \forall~N\trianglelefteq G$$ $$G~\text{is }F_2\Rightarrow G/N~\text{is }F_2 \forall~N\trianglelefteq G\text{ which are normally finitely generated}$$ My question is about the higher dimensional finiteness properties. Are there any similar statements about the quotient of an $F_n$ group being $F_n$?
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3$\begingroup$ I can't think of any nice "iff" condition like with finite presentability, but if the quotient is a retract or a quasi-retract, then it inherits type $F_n$, so that's a nice sufficient condition. See Theorem 8 of Alonso: core.ac.uk/download/pdf/82621017.pdf $\endgroup$– Matt ZaremskySep 27 at 19:56
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$\begingroup$ Not sure of the meaning of your last edit. Of course the reverse implication is true, but this is trivial (the more interesting "iff" is rather later in the sentence). Also the reverse implication is trivially true in the $F_1$ case. Multiple edits bump the question to the front page and this one wasn't that important. (By the way, it should spell "normally".) $\endgroup$– YCorSep 28 at 8:48
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$\begingroup$ @YCor Yep, sorry. I'll leave it as it was. Thanks $\endgroup$– MarcosSep 28 at 8:52
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