On the finiteness properties of fixed subgroups of automorphisms

TL;DR

This paper investigates the finiteness properties of fixed subgroups of automorphisms using Sigma-invariants, linking these properties to the group's center and quotient automorphisms, and addresses a specific open question.

Contribution

It introduces a novel approach using Sigma-invariants to analyze fixed subgroups' finiteness properties and provides an answer to an existing open question for groups with a direct factor center.

Findings

Established conditions relating fixed subgroup properties to the group's center and quotient automorphisms.

Provided a positive answer to Lei, Ma, and Zhang's question in the case of groups with a direct factor center.

Abstract

We use Sigma-invariants to study homotopical and homological finiteness properties of fixed subgroups of automorphisms of a group $G$ in terms of its center $Z(G)$ and the induced automorphisms on its associated quotient $G/Z(G)$. Specializing to the case where the center is a direct factor of the group, we answer a question made by Lei, Ma and Zhang.