# Does every f.g. group have a minimal presentation?

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.

• Look at arxiv.org/abs/1010.0271. I'm sure @YCor can say more. Jan 27 '20 at 18:07
• 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). Jan 27 '20 at 19:24
• Dear Benjamin: many thanks for providing the reference, it is spot on! Jan 27 '20 at 19:40
• Dear Luc: nice paper! I should have read it before asking! Jan 27 '20 at 19:41
• @AshotMinasyan, I believe that paper was one of the first talks I heard by YCor, and so I remembered it right away. Jan 27 '20 at 20:23

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