Call a group presentation $\langle X \,\|\,R \rangle$ minimal if no relator from $R$ is a consequence of the remaining relators, i.e., no $r \in R$ belongs to the normal closure of $R\setminus \{r\}$ in the free group $F(X)$.

Question: Does every finitely generated group have a minimal presentation (with $X$ finite)?

Remark: evidently every finite presentation $\langle X \,\|\,R \rangle$ has a minimal sub-presentation $\langle X \,\|\,R' \rangle$ (where $R' \subseteq R$ has the same normal closure in $F(X)$ as $R$), so the question really concerns infinitely presented groups.

It is possible to construct infinite presentations which do not have minimal sub-presentations. Indeed, let $F=F(a,b)$ be the free group on $\{a,b\}$. One can choose "sufficiently independent" elements $w_1,w_2,\dots \in F$ so that the presentation $$\langle a,b \,\|\, w_i^2,w^2_{i+1}w_i, ~i \in \mathbb{N} \rangle$$ has no minimal sub-presentation. Here $R$ consists of the elements $w_1^2,w_2^2,\dots$, and $w_2^2w_1,w_3^2w_2,\dots$.

However, the above group also has the presentation $\langle a,b \,\|\,w_1,w_2,\dots \rangle$, which may be minimal.

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    $\begingroup$ Look at arxiv.org/abs/1010.0271. I'm sure @YCor can say more. $\endgroup$ Jan 27 '20 at 18:07
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    $\begingroup$ Dear Ashot, the answer is negative: combine Theorem. 3.5 and Remark 5.3 of the paper cited by Benjamin Steinberg. A counter-example is given by the nilpotent-by-Abelian group $B$ of Equation (3.2). $\endgroup$
    – Luc Guyot
    Jan 27 '20 at 19:24
  • $\begingroup$ Dear Benjamin: many thanks for providing the reference, it is spot on! $\endgroup$ Jan 27 '20 at 19:40
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    $\begingroup$ Dear Luc: nice paper! I should have read it before asking! $\endgroup$ Jan 27 '20 at 19:41
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    $\begingroup$ @AshotMinasyan, I believe that paper was one of the first talks I heard by YCor, and so I remembered it right away. $\endgroup$ Jan 27 '20 at 20:23

The answer is no.

In order to see this, you may combine Theorem 3.9 and Remark 5.3 of [1]. A counter-example is given by the nilpotent-by-Abelian group $B$ of Equation (3.2). Further examples are provided by Remark 5.15.

[1] R. Bieri, Y. de Cornulier, L. Guyot and R. Strebel, "Infinite presentability of groups and condensation", 2014.


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