On the Bieri-Neumann-Strebel-Renz invariants of the weak commutativity construction \(\mathfrak{X}(G)\) (Q2694832)

Summary: 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.