Horospheres in Degenerate 3-Manifolds

Abstract

Abstract. We study horospheres in hyperbolic 3-manifolds M all whose ends are degenerate. Deciding which horospheres in M are properly embedded and which are dense reduces to a) studying the horospherical limit set; b) deciding which almost minimizing geodesics in M go through arbitrarily thin parts. As an answer to (a), we show that the horospherical limit set consists precisely of the injective points of the Cannon-Thurston map. Addressing (b), we provide characterizations, sufficient conditions as well as a number of examples and counterexamples.

Relations in bounded cohomology

“…The concatenation of these rays is a quasi-geodesic and tracks a geodesic closely; is uniformly thick, because M is. Moreover, is minimizing in the sense that there is a constant C > 0 such that d(p, (t)) |t| − C. This is because is essentially a limit of minimizing segments making linear progress away from p (see [21] for more about minimizing geodesics). Since the injectivity radius of M is bounded away from 0, and is minimizing, only comes within of itself finitely many times.…”

“…Recall that ∂ core(M n ) = Ψ n (S kn ). We use Inequalities (25) and (21) together with the triangle inequality to estimate…”

Cannon–thurston Maps

“…This will allow us to establish the existence of Cannon-Thurston maps. We shall focus on closed surfaces and follow the summary in [LM16] for the exposition.…”

Relations in Bounded Cohomology

“…The concatenation of these rays is a quasigeodesic and tracks a geodesic closely; is uniformly thick, because M is. Moreover, is minimizing in the sense that there is a constant C > 0 such that d(p, (t)) ≥ |t| − C. This is because is essentially a limit of minimizing segments making linear progress away from p (see [LM18] for more about minimizing geodesics). Since the injectivity radius of M is bounded away from 0, and is minimizing, only comes within of itself finitely many times.…”