Geometry & Topology

Hyperbolic Dehn filling in dimension four

Bruno Martelli and Stefano Riolo

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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.

Article information

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M50: Geometric structures on low-dimensional manifolds

hyperbolic $4$–manifolds cone manifolds Dehn filling


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.

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