Hyperbolic Dehn filling in dimension four

Martelli, Bruno; Riolo, Stefano

doi:10.2140/gt.2018.22.1647

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Hyperbolic Dehn filling in dimension four

Bruno Martelli and Stefano Riolo

Geometry & Topology 22 (2018) 1647–1716

DOI: 10.2140/gt.2018.22.1647

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 $M_{t}$ that interpolates between two hyperbolic four-manifolds $M_{0}$ and $M_{1}$ with the same volume $\frac{8}{3} π^{2}$ . 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.

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

Volume 22, issue 3 (2018)