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

Abstract

We compute the Bieri-Neumann-Strebel invariants Σ 1 for the generalized solvable Baumslag-Solitar groups Γ n and their finite index subgroups. Using Σ 1 , we show that certain finite index subgroups of Γ n cannot be isomorphic to Γ ñ for any ñ. In addition, we use the BNS-invariants to give a new proof of property R ∞ for the groups Γ n and their finite index subgroups.