We show that if all hyperbolic groups are residually finite, these statements follow: In relatively hyperbolic groups with peripheral structures consisting of finitely generated nilpotent subgroups, relatively quasiconvex subgroups are separable; geometrically finite subgroups of nonuniform lattices in rank one symmetric spaces are separable; Kleinian groups are subgroup separable. We also show that LERF for finite volume hyperbolic 3-manifolds would follow from LERF for closed hyperbolic 3-manifolds. We prove these facts by reducing, via combination and filling theorems, the separability of a relatively quasiconvex subgroup of a relatively hyperbolic group G to that of a quasiconvex subgroup of a hyperbolic quotientḠ. A result of Agol, Groves, and Manning is then applied. 1. Main results A subgroup H of a group G is called separable if for any g ∈ G \ H there is a homomorphism π onto a finite group such that π(g) ∈ π(H). A group is said to be residually finite if the trivial subgroup is separable, and subgroup separable or LERF if every finitely generated subgroup is separable. For example, Hall [1949] showed that free groups are LERF. It follows from a theorem of Mal'cev [1958] that polycyclic (and in particular finitely generated nilpotent) groups are LERF. A group is called slender if every subgroup is finitely generated. Polycyclic groups are also slender, by a result of Hirsch [1938]. Given a relatively hyperbolic group with peripheral structure consisting of LERF and slender subgroups, we study separability of relatively quasiconvex subgroups. This is connected, via filling constructions, to residual finiteness of hyperbolic groups. It is not known whether all word-hyperbolic groups are residually finite. Consequences of a positive or negative answer to this question have been explored by several authors; see for example [

We provide a new and elegant approach to relative quasiconvexity for relatively hyperbolic groups in the context of Bowditch's approach to relative hyperbolicity using cocompact actions on fine hyperbolic graphs. Our approach to quasiconvexity generalizes the other definitions in the literature that apply only for countable relatively hyperbolic groups. We also provide an elementary and self-contained proof that relatively quasiconvex subgroups are relatively hyperbolic. 20F06, 20F65, 20F67

We consider maximal slices of the Myers-Perry black hole, the doubly spinning black ring, and the Black Saturn solution. These slices are complete, asymptotically flat Riemannian manifolds with inner boundaries corresponding to black hole horizons. Although these spaces are simply connected as a consequence of topological censorship, they have non-trivial topology. In this note we investigate the question of whether the topology of spatial sections of the horizon uniquely determines the topology of the maximal slices. We show that the horizon determines the homological invariants of the slice under certain conditions. The homological analysis is extended to black holes for which explicit geometries are not yet known. We believe that these results could provide insights in the context of proving existence of deformations of this initial data. For the topological slices of the doubly spinning black ring and the Black Saturn we compute the homotopy groups up to dimension 3 and show that their 4-dimensional homotopy group is not trivial.

Let G be a group which is hyperbolic relative to a collection of subgroups H 1 , and it is also hyperbolic relative to a collection of subgroups H 2 . Suppose that H 1 ⊂ H 2 . We characterize when a relative quasiconvex subgroup of (G, H 2 ) is still relatively quasiconvex in (G, H 1 ). We also show that relative quasiconvexity is preserved when passing from (G, H 1 ) to (G, H 2 ). Applications are discussed.

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