English

Asymptotically geodesic hypersurfaces and the fundamental groups of hyperbolic manifolds

Geometric Topology 2026-03-27 v1 Differential Geometry

Abstract

We consider closed hypersurfaces smoothly immersed in hyperbolic manifolds up to homotopy and commensurability. We prove that if a closed hyperbolic manifold MM contains a sequence of asymptotically geodesic hypersurfaces, then π1(M)\pi_1(M) is virtually special and hence linear over integers. If MM (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 MM. This partially answers a question of Al Assal--Lowe. As a corollary, for each cocompact arithmetic lattice Γ\Gamma of SO(n+1,1)SO(n+1,1) of type I, there exist infinitely many arithmetic and infinitely many non-arithmetic cocompact lattices HH of SO(n,1)SO(n,1) that admit monomorphisms into Γ\Gamma which do not extend to a Lie group homomorphism from SO(n,1)SO(n,1) into SO(n+1,1)SO(n+1,1).

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Cite

@article{arxiv.2603.24869,
  title  = {Asymptotically geodesic hypersurfaces and the fundamental groups of hyperbolic manifolds},
  author = {Xiaolong Hans Han and Ruojing Jiang},
  journal= {arXiv preprint arXiv:2603.24869},
  year   = {2026}
}

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35 pages