On the Bieri-Neumann-Strebel-Renz invariants of the weak commutativity construction $X(G)$

For a finitely generated group $G$ we calculate the Bieri-Neumann-Strebel-Renz invariant $Sigma^1(X(G))$ for the weak commutativity construction $X(G)$. Identifying $S(X(G))$ with $S(X(G) / W(G))$ we show $Sigma^2(X(G),Z) subseteq Sigma^2(X(G)/ W(G),Z)$ and $Sigma^2(X(G)) subseteq $ $ Sigma^2(X(G)/ W(G))$ that are equalities when $W(G)$ is finitely generated and we explicitly calculate $Sigma^2(X(G)/ W(G),Z)$ and $ Sigma^2(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 $ u(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.