Volume 22, issue 3 (2018)

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Hyperbolic Dehn filling in dimension four

Bruno Martelli and Stefano Riolo

Geometry & Topology 22 (2018) 1647–1716
Abstract

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.

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