Decent actions of groups on restricted products

TL;DR

This paper investigates conditions under which group actions on restricted products are 'decent', establishing that automorphism groups of certain restricted products act decently, extending previous results.

Contribution

It introduces criteria for 'decent' actions on restricted products and proves that automorphism groups of such products over projective planes act decently.

Findings

Automorphism groups of restricted products over projective planes act decently.

Generalization of previous results on group actions on restricted products.

Provides new conditions for when group actions are 'decent'.

Abstract

An action of a group $G$ on a set $X$ is called ``decent'' if every subgroup of $G$ with a finite orbit in $X$ fixes a point in $X$ and every finitely generated subgroup of $G$ such that every element of the subgroup fixes a point of $X$ must itself have a global fixed point. In this article, we study conditions on when actions of groups on restricted products are ``decent''. We prove that the action of the automorphism group of a restricted product with base space the projective plane $\mathbb{P}^2(k)$ over a field $k$ is decent, generalizing a result of Lonjou--Przytycki--Urech.