The BNS invariants of the generalized solvable Baumslag-Solitar groups and of their finite index subgroups

Abstract

We compute the Bieri-Neumann-Strebel invariants ${\Sigma }^1$ for the generalized solvable Baumslag-Solitar groups ${\Gamma }_n$ and their finite index subgroups. Using ${\Sigma }^1$, we show that certain finite index subgroups of ${\Gamma }_n$ cannot be isomorphic to ${\Gamma }_k$ for any $k$. In addition, we use the BNS-invariants to give a new proof of property $R_{\infty }$ for the groups ${\Gamma }_n$ and their finite index subgroups.

20 pages