Decent actions of groups on restricted products

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.