Hyperbolic Dehn filling in dimension four.

*(English)*Zbl 1422.57045The authors construct the first explicit example of hyperbolic Dehn filling in dimension \(4\).

Hyperbolic Dehn filling is a characteristic phenomenon of \(3\)-dimensional hyperbolic geometry. One can start from a complete, finite-volume hyperbolic manifold and continuously deform the metric along a continuous path of non-complete hyperbolic metrics. The completion of these metrics contains a codimension-\(2\) singular locus, along which the total rotational angle is less than \(2\pi\) (a cone sigularity). When this value is precisely \(2\pi\), the metric completion is that of a new complete finite volume hyperbolic manifold, which is said to be obtained from the original one through hyperbolic Dehn filling.

The authors exhibit an example of an analogue phenomenon using \(4\)-dimensional hyperbolic manifolds instead. In this setting the topology and geometry is significantly more complicated. For instance, the singular locus of the completions consists of two surfaces (a torus and a Klein bottle) intersecting in two points, each surface with its own total rotational angle. Nonetheless, the initial and final points of the deformation consist of complete, finite volume hyperbolic manifolds which can be thought of as being obtained one from the other through \(4\)-dimensional hyperbolic Dehn filling.

The main ingredient of the construction is a path of continuous deformation of a non-compact hyperbolic \(4\)-dimensional polytope, the hyperbolic \(24\)-cell, which was originally discovered by S. P. Kerckhoff and P. A. Storm [Geom. Topol. 14, No. 3, 1383–1477 (2010; Zbl 1213.57023)]. The authors construct the corresponding manifolds with cone singularities by glueing together copies of these polytopes, and carefully analysing their topological and geometric structures.

Hyperbolic Dehn filling is a characteristic phenomenon of \(3\)-dimensional hyperbolic geometry. One can start from a complete, finite-volume hyperbolic manifold and continuously deform the metric along a continuous path of non-complete hyperbolic metrics. The completion of these metrics contains a codimension-\(2\) singular locus, along which the total rotational angle is less than \(2\pi\) (a cone sigularity). When this value is precisely \(2\pi\), the metric completion is that of a new complete finite volume hyperbolic manifold, which is said to be obtained from the original one through hyperbolic Dehn filling.

The authors exhibit an example of an analogue phenomenon using \(4\)-dimensional hyperbolic manifolds instead. In this setting the topology and geometry is significantly more complicated. For instance, the singular locus of the completions consists of two surfaces (a torus and a Klein bottle) intersecting in two points, each surface with its own total rotational angle. Nonetheless, the initial and final points of the deformation consist of complete, finite volume hyperbolic manifolds which can be thought of as being obtained one from the other through \(4\)-dimensional hyperbolic Dehn filling.

The main ingredient of the construction is a path of continuous deformation of a non-compact hyperbolic \(4\)-dimensional polytope, the hyperbolic \(24\)-cell, which was originally discovered by S. P. Kerckhoff and P. A. Storm [Geom. Topol. 14, No. 3, 1383–1477 (2010; Zbl 1213.57023)]. The authors construct the corresponding manifolds with cone singularities by glueing together copies of these polytopes, and carefully analysing their topological and geometric structures.

Reviewer: Leone Slavich (Pavia)

##### MSC:

57M50 | General geometric structures on low-dimensional manifolds |

57N16 | Geometric structures on manifolds of high or arbitrary dimension |

57K40 | General topology of 4-manifolds |

##### Citations:

Zbl 1213.57023
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\textit{B. Martelli} and \textit{S. Riolo}, Geom. Topol. 22, No. 3, 1647--1716 (2018; Zbl 1422.57045)

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