On the Bieri–Neumann–Strebel–Renz invariants of the weak commutativity construction $\mathfrak{X}(G)$

Abstract

For a finitely generated group $G$, we calculate the Bieri–Neumann–Strebel–Renz invariant $\Sigma^1(\mathfrak{X}(G))$ for the weak commutativity construction $\mathfrak{X}(G)$. Identifying $S(\mathfrak{X}(G))$ with $S(\mathfrak{X}(G) /\allowbreak W(G))$, we show $\Sigma^2(\mathfrak{X}(G),\mathbb{Z}) \subseteq \Sigma^2(\mathfrak{X}(G)/ W(G),\mathbb{Z})$ and $\Sigma^2(\mathfrak{X}(G)) \subseteq \Sigma^2(\mathfrak{X}(G)/ W(G))$, that are equalities when $W(G)$ is finitely generated, and we explicitly calculate $\Sigma^2(\mathfrak{X}(G)/ W(G),\mathbb{Z})$ and $\Sigma^2(\mathfrak{X}(G)/ W(G))$ in terms of the $\Sigma$-invariants of $G$. We calculate completely the $\Sigma$-invariants in dimensions 1 and 2 of the group $\nu(G)$ and show that if $G$ is finitely generated group with finitely presented commutator subgroup then the non-abelian tensor square $G \otimes G$ is finitely presented.