# zbMATH — the first resource for mathematics

Hyperbolic Dehn filling in dimension four. (English) Zbl 1422.57045
The 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.

##### 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
##### Keywords:
hyperbolic 4-manifolds; cone manifolds; Dehn filling
Zbl 1213.57023
Full Text:
##### References:
 [1] 10.1007/978-3-662-02901-5_1 [2] ; Andreev, Mat. Zametki, 8, 521, (1970) [3] 10.4007/annals.2005.162.195 · Zbl 1087.57009 [4] 10.1090/S0894-0347-04-00462-X · Zbl 1061.30037 [5] 10.2969/msjmemoirs/005010000 · Zbl 0955.57014 [6] ; Derevnin, Sibirsk. Mat. Zh., 45, 1022, (2004) [7] 10.4310/jdg/1287580965 · Zbl 1211.53066 [8] 10.2140/agt.2011.11.2237 · Zbl 1244.53051 [9] 10.2307/1970838 · Zbl 0206.03603 [10] 10.4310/jdg/1214460606 · Zbl 0919.57009 [11] 10.2140/gt.2010.14.1383 · Zbl 1213.57023 [12] 10.4310/jdg/1214461108 · Zbl 0990.57004 [13] 10.1007/s00039-013-0247-2 · Zbl 1283.57022 [14] 10.4171/GGD/147 · Zbl 1278.11047 [15] 10.4310/jdg/1312998235 · Zbl 1234.53014 [16] 10.1353/ajm.2017.0005 · Zbl 1364.53038 [17] 10.1016/j.crma.2005.03.019 · Zbl 1071.57020 [18] ; Mostow, Strong rigidity of locally symmetric spaces. Annals of Mathematics Studies, 78, (1973) · Zbl 0265.53039 [19] 10.1007/BF01418789 · Zbl 0264.22009 [20] 10.1090/proc/13272 · Zbl 1356.51012 [21] 10.2140/gtm.1998.1.511 [22] ; Vinberg, Uspekhi Mat. Nauk, 40, 29, (1985) [23] 10.4310/jdg/1143571990 · Zbl 1098.53038 [24] 10.4310/jdg/1180135696 · Zbl 1184.53049
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.