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Fix a group $G$ and fix a presentation of $G$ as $\langle X\mid R\rangle$. A presentationally finite extension of $G$ is any group that can be presented as $H=\langle X\cup X'\mid R\cup R'\rangle$, where $X',R'$ are finite. (Mind, the natural homomorphism $G\to H$ may not be injective. If necessary, we can restrict ourselves to cases where it is.)

For example, the presentationally finite extensions of the trivial group are the finitely presentable groups.

Question: Is there an existing name for this kind of group homomorphism/extension?

Unless I've missed something, it's not very hard to see that this does not depend on the choice of presentation of $G$ (sketch: adding extraneous generators to $X$ and the corresponding generation relations to $R$ does not change the isomorphism type (over $G$) of $\langle X\cup X'\mid R\cup R'\rangle$, neither does adding any relations that follow from the existing ones; thus, we can just add all elements of $G$ plus any number of their copies we want, and all relations true in $G$ about them and then remove any redundant ones we do not want to obtain all possible presentations).

Thus, this is really a property of the map $G\to H$ (or the extension $G\leq H$ if the map is injective) and not of a particular presentation.

I'm mostly interested in the case of finitely generated groups, but I think the idea works just as well for arbitrary groups, of arbitrary cardinality.

(Another way to describe this is simply as the groups of the form $H=(G*F)/N$, where $F$ is a free group of finite rank and $N$ is finitely generated as a normal subgroup of $G*F$, such that $N\cap G$ is trivial if we demand that the map $G\to H$ actually be injective.)

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    $\begingroup$ If you only add a single relation, this is called a “relative one-relator group”, which has some literature, eg here. $\endgroup$
    – Carl-Fredrik Nyberg Brodda
    Commented Feb 4, 2023 at 15:07
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    $\begingroup$ In analogy with the usage in algebraic geometry and commutative algebra, you could say that a morphism of groups $G\to H$ is "of finite presentation" if $H$ arises from $G$ by adding finitely many generators and relations, and call $H$ "finitely presented over $G$" $\endgroup$
    – Achim Krause
    Commented Feb 4, 2023 at 15:11
  • $\begingroup$ @AchimKrause: Oh, that's perfect. Thanks. If you post this as an answer, I will accept it. (Well, if there is some other terminology specific to groups, I will accept that, but I'm perfectly happy with this.) $\endgroup$
    – tomasz
    Commented Feb 4, 2023 at 15:14

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The kind of presentation you are giving is called a relative presentation and has been studied extensively by Steve Pride and his coauthors. So relatively finitely presented over $G$ is probably the best name (I didn't search his papers to see if he uses this term.)

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I'm not aware of any widely used term for this in the context of groups, but in analogy with the usage in algebraic geometry and commutative algebra, you could say that a morphism of groups $G\to H$ is "of finite presentation" if $H$ arises from $G$ by adding finitely many generators and relations, and call $H$ "finitely presented over $G$"

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  • $\begingroup$ I agree: I think that "finitely presented over $G$" is the right terminology (in general in universal algebra, hence for groups as well). $\endgroup$
    – YCor
    Commented Feb 4, 2023 at 15:47

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