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Asymptotically geodesic hypersurfaces and the fundamental groups of hyperbolic manifolds

Xiaolong Hans Han, Ruojing Jiang

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 $π_1(M)$ 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 $SO(n+1,1)$.

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 .

Paper Structure

This paper contains 22 sections, 44 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Outlines of proofs
  3. Some histories and motivations
  4. Preliminaries
  5. Arithmetic manifolds
  6. Tubular neighborhood of totally geodesic hypersurfaces
  7. Cube complex, virtual retract, and strong fillingness
  8. Basics
  9. Virtual retracts and homological injections
  10. Strongly filling subgroups and cubulations
  11. Nearly geodesic and asymptotically geodesic hypersurfaces
  12. Nearly geodesic hypersurfaces
  13. Asymptotically geodesic hypersurfaces are asymptotically dense in $\mathcal{G}_{n} M$
  14. Constructions
  15. The $\pi_1$-injectivity
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $\Gamma$ be a cocompact arithmetic lattice of $\textnormal{SO}(n+1,1)^{\circ}$ of type I and $M= \Gamma\backslash \mathbb{H}^{n+1}$. Then there exists a sequence $S_i$ of closed hyperbolic manifolds of dimension $n$ which admit $\pi_1$-injective immersions $\iota\colon S_i \hookrightarrow M$ suc Moreover, we can take $\pi_1(S_i)$ as arithmetic or non-arithmetic lattices of $\textnormal{SO}(n,1

Theorems & Definitions (77)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Definition 1.4: The principal curvature spectrum
  • Corollary 1.5
  • Proposition 1.6
  • Lemma 2.1: vsDiscreteGroupsConstantCurvature, p.221
  • Example 2.2: Infinitely many cocompact arithmetic lattices of $\textnormal{SO}(n,1)$ of type I
  • Remark 2.3
  • Theorem 2.4: baTubularNeighborhoodsTotallyGeodesic, Theorem 1.1
  • ...and 67 more