Relations between various boundaries of relatively hyperbolic groups

Suppose a group $G$ is relatively hyperbolic with respect to a collection $\text{PP}$ of its subgroups and also acts properly, cocompactly on a $\text{CAT}(0)$ (or $\delta$–hyperbolic) space $X$. The relatively hyperbolic structure provides a relative boundary $\mathrm{\partial }(G,\text{PP})$. The $\text{CAT}(0)$ structure provides a different boundary at infinity $\mathrm{\partial }X$. In this article, we examine the connection between these two spaces at infinity. In particular, we show that $\mathrm{\partial }(G,\text{PP})$ is $G$–equivariantly homeomorphic to the space obtained from $\mathrm{\partial }X$ by identifying the peripheral limit points of the same type.