Asymptotically geodesic hypersurfaces and the fundamental groups of hyperbolic manifolds

We consider closed hypersurfaces smoothly immersed in hyperbolic manifolds up to homotopy and commensurability. We prove that if a closed hyperbolic manifold $M$ contains a sequence of asymptotically geodesic hypersurfaces, then ${\pi }_1(M)$ is virtually special and hence linear over integers. If $M$ (dimension at least 3) is, in addition, arithmetic of type I, we constructs a sequence of hypersurfaces which are asymptotically geodesic (but not totally geodesic), strongly filling, and equidistributing in the Grassmann bundle over $M$. This partially answers a question of Al Assal–Lowe. As a corollary, for each cocompact arithmetic lattice $\mathrm{\Gamma }$ of $SO(n+1,1)$ of type I, there exist infinitely many arithmetic and infinitely many non-arithmetic cocompact lattices $H$ of $SO(n,1)$ that admit monomorphisms into $\mathrm{\Gamma }$ which do not extend to a Lie group homomorphism from $SO(n,1)$ into $SO(n+1,1)$.