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 $S(\mathfrak{X}(G)/W(G))$, 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.