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

TL;DR

This paper computes the Bieri-Neumann-Strebel-Renz invariants for the weak commutativity construction of a finitely generated group, revealing new relationships and explicit calculations in terms of the original group’s invariants.

Contribution

It provides explicit calculations of the $ ext{Sigma}$-invariants for the weak commutativity construction and relates them to the invariants of the original group $G$, including the non-abelian tensor square.

Findings

Calculated $ ext{Sigma}^1( ext{X}(G))$ for the weak commutativity construction.

Established equalities for $ ext{Sigma}^2( ext{X}(G))$ and $ ext{Sigma}^2( ext{X}(G)/W(G))$ under certain conditions.

Proved that if $G$ has a finitely presented commutator subgroup, then its non-abelian tensor square $G ensor G$ is finitely presented.

Abstract

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 $\nu(G)$ 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.