On the topology of conformally compact Einstein 4-manifolds

In this paper we study the topology of conformally compact Einstein 4-manifolds. When the conformal infinity has positive Yamabe invariant and the renormalized volume is also positive we show that the conformally compact Einstein 4-manifold will have at most finite fundamental group. Under the further assumption that the renormalized volume is relatively large, we conclude that the conformally compact Einstein 4-manifold is diffeomorphic to $B^4$ and its conformal infinity is diffeomorphic to $S^3$.