Postprint version. Published in Mathematische Annalen, Volume 372, Issue 3-4, December 1, 2018, pages 1103-1120.
The definitive version is available at https://doi.org/10.1007/s00208-018-1694-8.
In this paper we prove a conjecture of Alexander and Currier that states, except for covering maps of equidistant surfaces in hyperbolic 3-space, a complete, nonnegatively curved immersed hypersurface in hyperbolic space is necessarily properly embedded.
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