TITLE: Hyperbolic manifolds containing totally geodesic hypersurfaces which themselves contain totally geodesic hypersurfaces
QUESTION [5 upvotes]: I will preface this by saying that while I am familiar with the general theory of (semi)-Riemannian manifolds, I am a complete novice when it comes to the specifics of hyperbolic manifolds (I am using them for examples when computing cohomology with local coefficients). I was originally trained as a physicist, so I am also not very familiar with field theory (which comes up in the construction). So apologies in advance if there are some "dumb" questions here.
The purpose of my question is to understand the following construction referred to in the paper Local rigidity of hyperbolic manifolds with geodesic boundary by Kerckhoff and Storm, where they state:

"For each dimension $n>2$ [Gromov and Thurston] construct an infinite number of closed hyperbolic n-manifolds $V$ with the following properties: $V$ has a codimension 1 embedded, totally geodesic submanifold $M$ which itself has a codimension 1 embedded, totally geodesic submanifold $P$ (so $P$ is codimension 2 in $V$)."

The Gromov & Thurston paper being referred to is Pinching Constants for Hyperbolic Manifolds. It appears that the brief Section 1 contains the relevant construction, which I have snipped from the linked pdf:
I am unfortunately confused about "why" this construction gives us a hyperbolic manifold containing a codimension 1 totally geodesic hypersurface itself containing a totally geodesic codimension 1 hypersurface. To be more specific, why use this $\Phi_n$ instead of the usual Lorentzian quadratic form, and why finite index subgroups of this particular $\Gamma_n$?

REPLY [3 votes]: The fixed point set of the involution in the universal cover is a totally geodesic copy of hyperbolic space of codimension 1.  So that gives you the first totally geodesic hypersurface in some finite cover of the given thing.  And since the hypersurface is the same type of lattice, there is a finite cover of it that has a totally geodesic hypersurface by the same argument.  There is a single finite cover of the original thing doing both at once.