A note on the finitely generated fixed subgroup property

TL;DR

This paper investigates conditions under which groups of the form G×Z^m possess the finitely generated fixed subgroup property for automorphisms, utilizing the BNS-invariant to analyze partial cases and examples.

Contribution

It introduces new criteria for the finitely generated fixed subgroup property in groups G×Z^m using the BNS-invariant, along with partial results and examples.

Findings

Identifies conditions for FGFP_a in G×Z^m groups.

Provides partial answers to the FGFP_a problem.

Constructs non-trivial examples illustrating the property.

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.