Hyperbolic Dehn filling in dimension four

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 Mtthat interpolates between two hyperbolic four-manifolds M0and M1with the same volume 8/3π2. 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 2π. Here, the singularity of Mt is an immersed geodesic surface whose cone angles also vary monotonically from 0 to 2π. 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 2π, 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