Volume 22, issue 3 (2018)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 23
Issue 7, 3233–3749
Issue 6, 2701–3231
Issue 5, 2165–2700
Issue 4, 1621–2164
Issue 3, 1085–1619
Issue 2, 541–1084
Issue 1, 1–540

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Ethics Statement
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Author Index
To Appear
Other MSP Journals
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.

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