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Let a discrete group   $G$   act by homeomorphisms on compacta   $X$   and   $Y$   with convergence property. Suppose that the actions are minimal that is there is no non-empty proper invariant closed subsets. We will discuss the following question: whether there exist an action of   $G$   on a compactum   $Z$   with convergence property and continuous equivariant maps   $X\leftarrow Z\to Y$  ? We call the space   $Z$   (and action of   $G$   on it)  pullback  space (action). In such general setting, a negative answer follows from a recent result of Baker and Riley ( 1 ). Suppose, in addition, that the initial actions are relatively hyperbolic that is they are non-parabolic and the induced action on the distinct pairs are cocompact. In this case, the existence of the pullback space if   $G$   is finitely generated follows from ( 10 ). The main result of the paper is that the pullback space exists if and only if the maximal parabolic subgroups of one of the actions are dynamically quasiconvex for the other. We provide an example of two relatively hyperbolic actions of the free group   $G$   of countable rank for which the pullback action does not exist. We study an analog of the notion of geodesic flow for relatively hyperbolic groups. Further these results are used to prove the main theorem.

Similar Relatively Hyperbolic Actions of a Group