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.
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