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Consider a group presentation $ \left< G= \left\lbrace \text{generators}\right\rbrace \, \middle|\, R = \left\lbrace \text{relators}\right\rbrace \right>$ (no finiteness assumptions). Given $S\subset G$ let $R_S\subset R$ denote the subset made of all relators where only elements from $S$ are involved. The property that I consider is that for any $S\subset G$, the group with presentation $ \left< S \, \middle|\, R_S \right>$ is "naturally" a subgroup of the original group—that is, via the map that sends any generator to "itself".

Does this property have a name or has it already been studied in a particular context?

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    $\begingroup$ For one-relator groups this is Magnus's freiheitssatz. This property also holds for right angled artin groups. $\endgroup$
    – Benjamin Steinberg
    Mar 25, 2022 at 15:09
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    $\begingroup$ Note that this is a property of presentations and not of groups. This is true, as far as I know, for Coxeter and Artin presentations (as Benjamin Steinberg mentioned, of course with the given presentations). Note that some cases are trivial (e.g. even Artin/Coxeter presentations, i.e., where there are no finite odd labels — this encompasses right-angled Artin/Coxeter groups), since then killing a generator yields a retract. Also note that this is a property that is meaningful in universal algebra in general. $\endgroup$
    – YCor
    Mar 25, 2022 at 15:29
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    $\begingroup$ I think Freheitssatz (named after the famous theorem of Magnus mentioned by @BenjaminSteinberg) is the closest to a name that you will find for this. There have been many attempts to generalise the Freheitssatz in various ways, usually to various kinds of one-relator quotients -- see the oeuvre of Jim Howie and his collaborators... (tbc) $\endgroup$
    – HJRW
    Mar 25, 2022 at 18:03
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    $\begingroup$ ... Perhaps the most interesting example of such a theorem is Howie's "locally indicable Freiheitssatz". (Recall that a group $G$ is locally indicable if every non-trivial subgroup surjects $\mathbb{Z}$.) Howie proved that if $Y$ is a subcomplex of a 2-complex $X$, if $\pi_1Y$ is locally indicable, and if the relative second homology with rational coefficients $H_2(X,Y;\mathbb{Q})=0$ then the inclusion map $Y\to X$ is injective on $\pi_1$. $\endgroup$
    – HJRW
    Mar 25, 2022 at 18:05
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    $\begingroup$ I think this is related to Abels-Holz higher generation, as in (reader.elsevier.com/reader/sd/pii/…). More precisely this feels related to saying the group is "2-generated" by its subgroups generated by subsets of $S$. But off the top of my head I can't tell if this is equivalent. $\endgroup$
    – Matt Zaremsky
    Mar 26, 2022 at 10:20

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