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

TL;DR

This paper computes the Bieri-Neumann-Strebel invariants for generalized solvable Baumslag-Solitar groups and their finite index subgroups, revealing structural distinctions and providing new proofs of property R_infinity.

Contribution

It calculates the BNS invariants for these groups and uses them to distinguish subgroup isomorphisms and prove property R_infinity.

Findings

Certain finite index subgroups cannot be isomorphic to other groups of the same family.

The BNS invariants can be used to distinguish subgroup structures.

A new proof of property R_infinity for these groups is provided.

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.