Abstract
This paper initiates the study of rigidity for immersed, totally geodesic planes in hyperbolic 3-manifolds M of infinite volume. In the case of an acylindrical 3-manifold whose convex core has totally geodesic boundary, we show that the closure of any immersed geodesic plane is a properly immersed submanifold of M. On the other hand, we show that rigidity fails for quasifuchsian manifolds.
Highlights
Let M = Γ\H3 be an oriented complete hyperbolic 3-manifold, presented as the quotient of hyperbolic space by the action of a Kleinian groupΓ ⊂ G = PGL2(C) ∼= Isom+(H3).Let f : H2 → M be a geodesic plane, i.e. a totally geodesic immersion of a hyperbolic plane into M .We often identify a geodesic plane with its image, P = f (H2)
To give the proof of Theorem 1.5, we first develop some general results on Kleinian groups which may be of independent interest
Since hull(E) ⊂ hull(Λ), this means that the limiting frame y must be tangent to a geodesic plane entirely contained in the convex core of M
Summary
The proof of Theorem 1.1 uses, in part, an isolation theorem for compact geodesic surfaces, which leads to the following finiteness result: Theorem 1.4 Let M = Γ\H3 be an infinite volume hyperbolic 3-manifold with compact convex core. Since γ can be a fairly wild set (e.g. locally a Cantor set times [0, 1]), the closure of a geodesic plane in a Fuchsian manifold certainly need not be a submanifold, in contrast to Corollary 1.2 This type of example is robust, in the sense that it persists for quasifuchsian groups (see Appendix A). This result is needed to show that C ∩ Λ is a Cantor set in cases (2) and (3) of Theorem 1.5. We would like to thank Ian Agol for raising the question of the measure of E in Theorem 6.1
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