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 that interpolates between two hyperbolic four-manifolds and with the same volume . 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 to . Here, the singularity of is an immersed geodesic surface whose cone angles also vary monotonically from to . When a cone angle tends to 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 , 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.
Citation
Bruno Martelli. Stefano Riolo. "Hyperbolic Dehn filling in dimension four." Geom. Topol. 22 (3) 1647 - 1716, 2018. https://doi.org/10.2140/gt.2018.22.1647
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