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Hyperbolic Dehn filling in dimension four

Bruno Martelli and Stefano Riolo

Geometry & Topology 22 (2018) 1647–1716

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 Mt that interpolates between two hyperbolic four-manifolds M0 and M1 with 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.

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hyperbolic $4$–manifolds, cone manifolds, Dehn filling
Mathematical Subject Classification 2010
Primary: 57M50
Received: 29 September 2016
Accepted: 26 July 2017
Published: 16 March 2018
Proposed: Benson Farb
Seconded: Anna Wienhard, András I Stipsicz
Bruno Martelli
Dipartimento di Matematica
Università di Pisa
Stefano Riolo
Dipartimento di Matematica
Università di Pisa