Asymptotically geodesic hypersurfaces and the fundamental groups of hyperbolic manifolds
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| Format: | Preprint |
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2026
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| _version_ | 1866912982889398272 |
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| author | Han, Xiaolong Hans Jiang, Ruojing |
| author_facet | Han, Xiaolong Hans Jiang, Ruojing |
| contents | 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 $π_1(M)$ 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 $Γ$ 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 $Γ$ which do not extend to a Lie group homomorphism from $SO(n,1)$ into $SO(n+1,1)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_24869 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Asymptotically geodesic hypersurfaces and the fundamental groups of hyperbolic manifolds Han, Xiaolong Hans Jiang, Ruojing Geometric Topology Differential Geometry 57K32, 53C42, 20F67 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 $π_1(M)$ 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 $Γ$ 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 $Γ$ which do not extend to a Lie group homomorphism from $SO(n,1)$ into $SO(n+1,1)$. |
| title | Asymptotically geodesic hypersurfaces and the fundamental groups of hyperbolic manifolds |
| topic | Geometric Topology Differential Geometry 57K32, 53C42, 20F67 |
| url | https://arxiv.org/abs/2603.24869 |