On the Bieri-Neumann-Strebel-Renz invariants and limit groups over Droms RAAGs

TL;DR

This paper investigates the Bieri-Neumann-Strebel-Renz invariants for certain limit groups over Droms RAAGs, revealing conditions under which these invariants are empty or computable, advancing understanding of their algebraic properties.

Contribution

It provides new calculations and conditions for the BNSR invariants of limit groups over Droms RAAGs and finitely presented residually Droms RAAGs, extending previous theoretical frameworks.

Findings

For limit groups over Droms RAAGs with trivial center, (G) and (G, \u211d) are empty.

Calculated (H) and (H)_{dis} for finitely presented residually Droms RAAGs.

Established a necessary condition for  to belong to (H).

Abstract

For a group $G$ that is a limit group over Droms RAAGs such that $G$ has trivial center, we show that $\Sigma^1(G) = \emptyset = \Sigma^1(G, \mathbb{Q})$. For a group $H$ that is a finitely presented residually Droms RAAG we calculate $\Sigma^1(H)$ and $\Sigma^2(H)_{dis}$. In addition, we obtain a necessary condition for $[\chi]$ to belong to $\Sigma^n(H)$.