Let $F\_n$ be a free group of rank $n \geq 2$. The group $F\_n$ acts on its commutator subgroup $[F\_n,\, F\_n]$ by conjugation. Let $G = [F\_n,\, F\_n] \rtimes F\_n$. It's not hard to see that $G$ is finitely generated.



> 
> **Question**: Is $G$ finitely presentable? Presumably not, but I can't seem to prove it.
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Finite presentability of semi-direct product of free group and its commutator subgroup
This group is not finitely presentable. Indeed, write $F$ for the given free group and $F'$ for its derived subgroup. The map
$$F\ltimes F'\to F\times F,\quad (f,g)\mapsto (f,fg)$$
is an injective group homomorphism, and its image is the fibre product of two copies of $F$ over the abelianization map $F\to F/F'$. (Alternatively, start from this fibre product, and observe that it is the semidirect product $H\ltimes K$, where $H\simeq F$ is the diagonal and $K=F'\times\{1\}$.)


It is a result of Baumslag and Roseblade that a subgroup of a direct product of two free groups is "almost never" finitely presented, i.e. not finitely presentable unless one of the projection is injective on the subgroup, or if it is virtually the direct product of intersection with the factors. Since this is not such an exception, the above group is not finitely presentable.


Reference: G. Baumslag and J. E. Roseblade, Subgroups of direct products of free groups, J. London Math. Soc. (2) 30 (1984), 44-52. [link](https://watermark.silverchair.com/s2-30-1-44.pdf?token=AQECAHi208BE49Ooan9kkhW_Ercy7Dm3ZL_9Cf3qfKAc485ysgAAAs4wggLKBgkqhkiG9w0BBwagggK7MIICtwIBADCCArAGCSqGSIb3DQEHATAeBglghkgBZQMEAS4wEQQMwl7Az8Gm-1Aoh0m6AgEQgIICgXL9-Y-fyUaMyul-VSo28p2ygbYHC3OaDDC0tZUL7PTcTHZc0-M_IIjukqs47Fxkhln44ybAh97Y7-QpifPTc0qNqoq5OVpFYyp3Klok_kGXnHS98io2mtVb-b6uXrGrQmnG5uEZVoZrNNQ8m9P5gDbXV2oCuBul6GE6StDbM9nPlOch-r8Bd4GF9_iUWH6UN1ILHT-NvnjhTdhdVblGu7hnULKGq1Jf_U30vCXGAYpNM_xdLN-QznjXjRpMNlgjPlY8-kCtX1RDVSPjqQiCUb8Q8AMDIxRh2-qWIKnZdYD-1djeYYmPz52KQ01eKmdaivOjZDauzeQWMVBGAwPy7b_GA1gOiuOTlHDC3avPA-BTXjwA00s7xsZIDm4AEPbEyD1Sh6zRaYacGDLjlUqQhFm14CunJkO_vYxepGI3o945EQ6InYOmzwMejo9ZplDhPknjCHVAANkVSQCBaBp0PxkwuKI2rq8PRFjEpndzk1VhRQdB91vh9UDiiUUJ1eRkmUO7RfAmpDKOtx37rJMuHb8LgcB5yt2QMizOjhvBhPadamlvlGB5PpqPg-MGQANOK4tzGl5vZTxBPx8YY45dLlw79kyp0vqXm3YwJmRW14oR77rN-h3mluTzzDuVGxl-1TyN23TlH_19trKximYz5dYu1PXG1DLlkAg-vQDEdySLci_gCd7cZ88ZKLh0-8ENJSg9ynlas-WTUO4wkjIDTRmd4Vz9_7KXw9ixUXLFjKFZxkBMDSNpgBiMhXYjob3AMus-QVNwY471ijwFIe2HwVkRQPUia5ZBh51-cwhi_W3FI98Z2DRhcg0XtybqRVfpZpK-xA8K744jVi5Ekl3dATJX) [DOI behind paywall](https://doi.org/10.1112/jlms/s2-30.1.44)


PS: the same argument works equally if $N$ is an arbitrary nontrivial normal subgroup of infinite index in $F$: then $F\ltimes N$ is not finitely presentable.


