In this talk, we will explain how the quasi-conformal deformation spaces of kissing reflection groups can be used to describe the moduli space of circle packings. We will compare them with the hyperbolic components for critically fixed anti-rational maps. We will explain how the analogue of Thurston's compactness theorem for acylindrical hyperbolic 3-manifold holds for critically fixed anti-rational maps. We will further analyze the bumping structures for those hyperbolic components for critically fixed anti-rational maps, proving an analogue of an unpublished result by Hatcher and Thurston for the moduli space of circle packings. Speaker(s): Yusheng Luo (U(M))