Hyperbolic Dehn filling in dimension four

TL;DR

This paper extends the concept of hyperbolic Dehn filling to four-dimensional hyperbolic manifolds, constructing a deformation path between two such manifolds with the same volume, involving cone angles and singularities.

Contribution

It introduces a novel four-dimensional analogue of Thurston's hyperbolic Dehn filling, utilizing deforming hyperbolic polytopes to construct explicit deformation paths.

Findings

Constructed an analytic path of cone four-manifolds interpolating between two hyperbolic four-manifolds.

Demonstrated how cone angles vary monotonically from 0 to 2π, creating different geometric structures.

Showed instances of hyperbolic Dehn fillings and degenerations in four dimensions.

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.