Geometry & Topology
- Geom. Topol.
- Volume 22, Number 3 (2018), 1647-1716.
Hyperbolic Dehn filling in dimension four
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.
Geom. Topol., Volume 22, Number 3 (2018), 1647-1716.
Received: 29 September 2016
Accepted: 26 July 2017
First available in Project Euclid: 31 March 2018
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 57M50: Geometric structures on low-dimensional manifolds
Martelli, Bruno; Riolo, Stefano. Hyperbolic Dehn filling in dimension four. Geom. Topol. 22 (2018), no. 3, 1647--1716. doi:10.2140/gt.2018.22.1647. https://projecteuclid.org/euclid.gt/1522461625