Equivariant blowups of bounded parabolic points

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.