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](https://mathoverflow.net/questions/312548/hyperbolic-manifolds-containing-totally-geodesic-hypersurfaces-which-themselves) 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!