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