Relations between various boundaries of relatively hyperbolic groups

Suppose a group $G$ is relatively hyperbolic with respect to a collection \PP of its subgroups and also acts properly, cocompactly on a \CAT(0) (or $\delta$--hyperbolic) space $X$. The relatively hyperbolic structure provides a relative boundary \partial(G,\PP). The \CAT(0) structure provides a different boundary at infinity $\partial X$. In this article, we examine the connection between these two spaces at infinity. In particular, we show that \partial (G,\PP) is $G$--equivariantly homeomorphic to the space obtained from $\partial X$ by identifying the peripheral limit points of the same type.Comment: 22 page