Calculations with relation modules

Question

Suppose we have a finitely presented group $G$ with free presentation $$ R\hookrightarrow F \twoheadrightarrow G. $$ To this presentation we may associate an extension with abelian kernel $$ R/R' \hookrightarrow F/R' \twoheadrightarrow G. $$ Here $R'$ denotes the commutator subgroup of $R$, and the $G$-module $R/R'$ is sometimes known as the relation module associated to the presentation.

Under what conditions is the group $F/R'$ finitely presentable?

To give a concrete example, consider the free product of cyclic groups $C_2\ast C_3$ with presentation $\langle a,b \mid a^2,b^3 \rangle$.

Is there a nice finite presentation of $F/R'$ in this case?

I would also be interested in any pointers to places in the literature where calculations for particular presentations are carried out.

Answer 1

Let $F$ be a finitely generated non-abelian free group with a non-trivial normal subgroup $R \lhd F$. Suppose that $G:=F/R$ is finitely presented. Then the group $F/R'$ is finitely presented if and only if $G$ is finite. In particular, if $G=C_2*C_3$ then $F/R'$ is not finitely presented.

This is mentioned in the paper of Kulikova and Ol'shanskii (MR2523056 Kulikova, O. V.; Olʹshanskiĭ, A. Yu. "On the finite presentability of the group $F/[M,N]$." (Russian. Russian summary) Vestnik Moskov. Univ. Ser. I Mat. Mekh. 2006, no. 6, 19--21, 62; translation in Moscow Univ. Math. Bull. 61 (2006), no. 6, 20–23 (2007); http://arxiv.org/abs/math/0703604) and is attributed to Shmel'kin (MR0193131 Šmelʹkin, A. L. "Wreath products and varieties of groups." (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 29 1965 149–170).