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

Abstract

For a finitely generated group G we calculate the Bieri-Neumann-) that are equalities when W (G) is finitely generated and we explicitly calculate Σ 2 (X(G)/W (G), Z) and Σ 2 (X(G)/W (G)) in terms of the Σ-invariants of G. We calculate completely the Σ-invariants in dimensions 1 and 2 of the group ν(G) and show that if G is finitely generated group with finitely presented commutator subgroup then the non-abelian tensor square G ⊗ G is finitely presented.