Do you know how to construct a compact hyperbolic 3-manifold with three or four totally geodesic boundary components? The only constructions I could find have one boundary component. A reference would be appreciated.



Do you know how to construct a compact hyperbolic 3-manifold with three or four totally geodesic boundary components?
**Theorem**([Long and Niblo](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=AUCN&pg6=PC&pg7=ALLF&pg8=ET&r=1&review_format=html&s4=long&s5=niblo&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq)): If $M$ is a 3-manifold and $S$ is an incompressible component of $\partial M$ then $\pi\_1 S$ is *separable* in $\pi\_1 M$ (pick a base point in $S$ to make sense of this).


This means that for any $\gamma\in \pi\_1M\smallsetminus \pi\_1 S$ there is a homomorphism $f$ from $\pi\_1M$ to a finite group such that $f(\gamma)\notin f(\pi\_1(S))$.


**Corollary:** If $|\pi\_1M:\pi\_1S|=\infty$ then $M$ had finite-sheeted coverings such that the preimage of $S$ has as many components as you like.


**Proof:** Choose a homomorphism $f$ such that $f(\pi\_1S)$ has large index in $f(\pi\_1M)$. Now your covering corresponds to $\ker f$. **QED** 


So examples exists as finite covers of the example you already have!


