标题： 关于弱交换构造的Bieri-Neumann-Strebel-Renz不变量 $\X(G)$ 显示英文标题

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

摘要： 对于一个有限生成群$G$，我们计算弱交换构造$\X(G)$的 Bieri-Neumann-Strebel-Renz 不变量$\Sigma^1(\X(G))$。 将$S(\X(G))$与$S(\X(G) / W(G))$联立，我们证明了$\Sigma^2(\X(G),\Z) \subseteq \Sigma^2(\X(G)/ W(G),\Z)$和$\Sigma^2(\X(G)) \subseteq $ $ \Sigma^2(\X(G)/ W(G))$ ，当$W(G)$是有限生成时它们是等式，并我们显式计算了$\Sigma^2(\X(G)/ W(G),\Z)$和$ \Sigma^2(\X(G)/ W(G))$，它们是基于$\Sigma$的不变量的$G$。 我们完全计算了群$\nu(G)$在维度 1 和 2 的$\Sigma$-不变量，并证明如果$G$是一个有限生成群且换位子群是有限表示的，则非交换张量平方$G \otimes G$是有限表示的。 显示英文摘要

摘要： 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.