I am thinking about the question that: if we double a hyperbolic 3 manifold along its boundary, will the rank of fundemental group of the resulting closed manifold be strictly larger than before?\ The answer is "Yes" if the following question is true. Does every hyperbolic 3 manifold with totally geodesic boundary has some connected finite covering space with more than one boundary component?
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10$\begingroup$ This follows from peripheral separability of incompressible surfaces in the boundary of a compact 3-manifold: ams.org/mathscinet-getitem?mr=1109662 $\endgroup$– Ian AgolCommented Oct 2, 2011 at 15:44
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$\begingroup$ A slightly older reference which applies in your case is a result of Long: ams.org/mathscinet-getitem?mr=898729 Also to clarify, these techniques only apply in the case of finite-volume manifolds with totally geodesic boundary: I'm not sure what is known in the case of infinitely generated groups. $\endgroup$– Ian AgolCommented Oct 3, 2011 at 0:48
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$\begingroup$ Agol: I think your comment would make a good answer. $\endgroup$– S. Carnahan ♦Commented Oct 4, 2011 at 8:27
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