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I am hoping to find examples of compact hyperbolic manifolds with at least 2 disjoint totally geodesic hypersurfaces. Ideally, I would like examples in dimension at least 4, though 3-dimensional manifolds would be good too. At least one of these hypersurfaces should be separating (the other(s) need not be).

I know that arithmetic techniques can be used to construct hyperbolic manifolds containing a totally geodesic hypersurface which is separating (and can be adapted to get nested totally geodesic submanifolds). However, I am not experienced enough with hyperbolic geometry/geometric group theory/number theory to know if these techniques can be adapted to create arithmetic hyperbolic manifolds which contain disjoint totally geodesic hypersurfaces.

Any references would be greatly appreciated!

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I do not have a self-contained reference, but the key is

Long, D. D.; Reid, A. W., Constructing hyperbolic manifolds which bound geometrically, Math. Res. Lett. 8, No. 4, 443-455 (2001). ZBL0992.57023.

where they construct, for every $n\ge 2$ a closed hyperbolic $n$-manifold $N$ and a compact separating connected totally geodesic hypersurface $M\subset N$. By adapting their proof, one obtains examples of manifolds $N$ containing not only manifolds $M$ as above but arbitrariliy large (finite) number of non-separating compact totally-geodesic hypersurfaces disjoint from $M$. But to prove this, one has to understand the details of the proof of their main theorem. In dimension 3 one can give a different proof using Thurston's hyperbolization theorem.

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    $\begingroup$ 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 $\endgroup$
    – Ian Agol
    Commented Nov 19, 2023 at 2:49
  • $\begingroup$ 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. $\endgroup$
    – ಠ_ಠ
    Commented Nov 19, 2023 at 3:56

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