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2018 Hyperbolic Dehn filling in dimension four
Bruno Martelli, Stefano Riolo
Geom. Topol. 22(3): 1647-1716 (2018). DOI: 10.2140/gt.2018.22.1647


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 M t that interpolates between two hyperbolic four-manifolds M 0 and M 1 with the same volume 8 3 π 2 . 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 2 π . Here, the singularity of M t is an immersed geodesic surface whose cone angles also vary monotonically from 0 to  2 π . 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 2 π , 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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Bruno Martelli. Stefano Riolo. "Hyperbolic Dehn filling in dimension four." Geom. Topol. 22 (3) 1647 - 1716, 2018.


Received: 29 September 2016; Accepted: 26 July 2017; Published: 2018
First available in Project Euclid: 31 March 2018

zbMATH: 06864265
MathSciNet: MR3780443
Digital Object Identifier: 10.2140/gt.2018.22.1647

Primary: 57M50

Rights: Copyright © 2018 Mathematical Sciences Publishers


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Vol.22 • No. 3 • 2018
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