Asymptotically geodesic hypersurfaces and the fundamental groups of hyperbolic manifolds
Abstract
We consider closed hypersurfaces smoothly immersed in hyperbolic manifolds up to homotopy and commensurability. We prove that if a closed hyperbolic manifold contains a sequence of asymptotically geodesic hypersurfaces, then is virtually special and hence linear over integers. If (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 . This partially answers a question of Al Assal--Lowe. As a corollary, for each cocompact arithmetic lattice of of type I, there exist infinitely many arithmetic and infinitely many non-arithmetic cocompact lattices of that admit monomorphisms into which do not extend to a Lie group homomorphism from into .
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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