I am hoping to find examples of compact hyperbolic manifolds with 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 reference requests would be greatly appreciated!

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!