A note on the finitely generated fixed subgroup property

TL;DR

This work investigates when a direct product $G\times \mathbb{Z}^m$ has the finitely generated fixed subgroup property $\mathrm{FGFP}_a$ for automorphisms, employing the BNS-invariant $\Sigma^1(G)$ as the central tool. It derives necessary conditions, notably that $G$ must have $\mathrm{FGFP}_a$ and that $S\mathbb{Q}(G)\subset \Sigma^1(G)$, and identifies when $\Sigma^1(G)=S(G)$, linking to the finite generation of $[G,G]$. It also provides sufficient conditions, including centerless $G$ with $\mathrm{FGFP}_a$ and $S\mathbb{Q}(G)\subset \Sigma^1(G)$, under which $G\times \mathbb{Z}^m$ has $\mathrm{FGFP}_a$ for all $m\ge1$, with additional results when $[G,G]$ is finitely generated or $\Sigma^1(G)=S(G)$. The paper culminates in non-trivial examples, notably slender groups and certain hyperbolic 3-manifold groups, demonstrating $\mathrm{FGFP}_a$ for all $m$, and discusses the scope and limitations of these conditions in broader classes of groups.

Abstract

We study when a group of form $G\times\mathbb{Z}^m (m\geq 1)$ has the finitely generated fixed subgroup property of automorphisms ($\rm{FGFP}_a$), by using the BNS-invariant, and provide some partial answers and non-trivial examples.

Theorems & Definitions (22)

• Example 1.1
• Definition 2.1
• Theorem 2.2: Bieri-Neumann-Strebel
• Proposition 2.3
• proof
• Theorem 3.1
• proof
• Remark 3.2
• Example 3.3
• Definition 3.4