Fixed subgroups in direct products of surface groups of Euclidean type

TL;DR

This paper characterizes when fixed subgroups of automorphisms in direct products of Euclidean-type surface groups are compressed or inert, providing explicit criteria for these subgroup properties.

Contribution

It offers an explicit characterization of fixed subgroup properties in direct products of Euclidean surface groups, a novel result in geometric group theory.

Findings

Fixed subgroups are compressed under certain conditions.

Fixed subgroups are inert in specific cases.

Provides explicit criteria for subgroup properties.

Abstract

We give an explicit characterization of which direct products $G$ of surface groups of Euclidean type satisfy that the fixed subgroup of any automorphism (or endomorphism) of $G$ is compressed, and of which is it always inert.