Equivariant blowups of bounded parabolic points

TL;DR

This paper introduces a method to modify group actions on compact spaces by equivariantly blowing up bounded parabolic points, enabling new insights into convergence actions and properties of stabilizers.

Contribution

It constructs a new space with an equivariant blowup of bounded parabolic points, facilitating the analysis and creation of convergence actions of groups.

Findings

Characterizes spaces with convergence group actions

Constructs new convergence actions from existing ones

Shows stabilizers of bounded parabolic points are one-ended

Abstract

Let $G$ be a group acting by homeomorphisms on a Hausdorff compact space $Z$. We constructed a new space $X$ that blows up equivariantly the bounded parabolic points of $Z$. This means, roughly speaking, that $G$ acts by homeomorphisms on $X$ and there exists a continuous equivariant map $\pi: X \rightarrow Z$ such that for every non bounded parabolic point $z \in Z$, $\#\pi^{-1}(z) = 1$. We use such construction to characterize topologically some spaces that $G$ acts with the convergence property and to construct new convergence actions of $G$ from old ones. As one of the applications, if $G$ is a group and $p$ is a bounded parabolic point of the space of ends of $G$, then the stabilizer of $p$ is one-ended.