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Nov 19, 2023 at 3:56 comment added ಠ_ಠ Thank you very much both of you! I had actually read the Long and Reid paper a couple years ago but I did not understand their techniques well enough to adapt them.
Nov 19, 2023 at 3:53 vote accept ಠ_ಠ
Nov 19, 2023 at 2:49 comment added Ian Agol Once you have a manifold $M$ with a (connected) geodesic boundary $W=\partial M$, the fundamental group of $W$ is separable *, and hence it lifts to a (non-trivial) finite-sheeted cover $M’$ of $M$. Double $M’$ along $\partial M’-W$, then cap off the two copies of $W$ with two copies of $M$. This should give a desired manifold. * separability by Long mathscinet.ams.org/mathscinet/article?mr=898729
Nov 17, 2023 at 3:40 history answered Moishe Kohan CC BY-SA 4.0