Ubiquitous quasi-Fuchsian surfaces in cusped hyperbolic 3–manifolds

Cooper, Daryl¹ ; Futer, David²

¹ Department of Mathematics, University of California at Santa Barbara, Santa Barbara, CA, United States
² Department of Mathematics, Temple University, Philadelphia, PA, United States

Geometry & topology, Tome 23 (2019) no. 1, pp. 241-298.

Résumé

We prove that every finite-volume hyperbolic $3$–manifold $M$ contains a ubiquitous collection of closed, immersed, quasi-Fuchsian surfaces. These surfaces are ubiquitous in the sense that their preimages in the universal cover separate any pair of disjoint, nonasymptotic geodesic planes. The proof relies in a crucial way on the corresponding theorem of Kahn and Markovic for closed $3$–manifolds. As a corollary of this result and a companion statement about surfaces with cusps, we recover Wise’s theorem that the fundamental group of $M$ acts freely and cocompactly on a $CAT\left(0\right)$ cube complex.

Détail

DOI : 10.2140/gt.2019.23.241

Classification : 20F65, 20H10, 30F40, 57M50
Keywords: hyperbolic 3-manifold, immersed surface, quasifuchsian, cubulation

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