Hyperbolic Dehn filling in dimension four

Martelli, Bruno¹ ; Riolo, Stefano¹

¹ Dipartimento di Matematica, Università di Pisa, Pisa, Italy

Geometry & topology, Tome 22 (2018) no. 3, pp. 1647-1716.

Résumé

We introduce and study some deformations of complete finite-volume hyperbolic four-manifolds that may be interpreted as four-dimensional analogues of Thurston’s hyperbolic Dehn filling.

We construct in particular an analytic path of complete, finite-volume cone four-manifolds $M_t$ that interpolates between two hyperbolic four-manifolds $M_0$ and $M_1$ with the same volume $\frac{8}{3}{\pi }^2$. The deformation looks like the familiar hyperbolic Dehn filling paths that occur in dimension three, where the cone angle of a core simple closed geodesic varies monotonically from $0$ to $2\pi$. Here, the singularity of $M_t$ is an immersed geodesic surface whose cone angles also vary monotonically from $0$ to $2\pi$. When a cone angle tends to $0$ a small core surface (a torus or Klein bottle) is drilled, producing a new cusp.

We show that various instances of hyperbolic Dehn fillings may arise, including one case where a degeneration occurs when the cone angles tend to $2\pi$, like in the famous figure-eight knot complement example.

The construction makes an essential use of a family of four-dimensional deforming hyperbolic polytopes recently discovered by Kerckhoff and Storm.

Détail

DOI : 10.2140/gt.2018.22.1647

Classification : 57M50
Keywords: hyperbolic $4$–manifolds, cone manifolds, Dehn filling

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