Asymptotically geodesic hypersurfaces and the fundamental groups of hyperbolic manifolds

TL;DR

This paper investigates the properties of asymptotically geodesic hypersurfaces in hyperbolic manifolds, demonstrating their implications for the fundamental groups' structure and constructing sequences with specific geometric and algebraic properties.

Contribution

It proves that the presence of asymptotically geodesic hypersurfaces implies the fundamental group is virtually special and constructs such hypersurfaces in arithmetic hyperbolic manifolds, partially answering an open question.

Findings

Fundamental groups are virtually special and linear over integers.

Existence of sequences of asymptotically geodesic, strongly filling hypersurfaces in arithmetic manifolds.

Construction of non-extension monomorphisms between lattices in different orthogonal groups.

Abstract

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 $\Gamma$ 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 $\Gamma$ which do not extend to a Lie group homomorphism from $SO(n,1)$ into $SO(n+1,1)$.