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Surgeries of the Gieseking hyperbolic ideal simplex manifold
- Publication Year :
- 2020
- Publisher :
- arXiv, 2020.
-
Abstract
- In our Novi Sad conference paper (1999) we described Dehn type surgeries of the famous Gieseking (1912) hyperbolic ideal simplex manifold $\mathcal{S}$, leading to compact fundamental domain $\mathcal{S}(k)$, $k = 2, 3, \dots$ with singularity geodesics of rotation order $k$, but as later turned out with cone angle $2(k-1)/k$. We computed also the volume of $S(k)$, tending to zero if $k$ goes to infinity. That time we naively thought that we obtained orbifolds with the above surprising property. As the reviewer of Math. Rev., Kevin P. Scannell (MR1770996 (2001g:57030)) rightly remarked, "this is in conflict with the well-known theorem of D. A. Kazhdan and G. A. Margulis (1968) and with the work of Thurston, describing the geometric convergence of orbifolds under large Dehn fillings". In this paper we refresh our previous publication. Correctly, we obtained cone manifolds (for $k > 2$), as A. D. Mednykh and V. S. Petrov (2006) kindly pointed out. We complete our discussion and derive the above cone manifold series (Gies.1 and Gies.2) in two geometrically equivalent form, by the half turn symmetry of any ideal simplex. Moreover we obtain a second orbifold series (Gies.3 and 4), tending to the regular ideal simplex as the original Gieseking manifold.<br />19 pages, 9 figures
- Subjects :
- Simplex
Ideal (set theory)
General Mathematics
Order (ring theory)
Geometric Topology (math.GT)
Metric Geometry (math.MG)
Type (model theory)
Mathematics::Geometric Topology
Manifold
Combinatorics
Mathematics - Geometric Topology
57M50, 57N10
Mathematics - Metric Geometry
Gieseking manifold
FOS: Mathematics
Regular ideal
Orbifold
Mathematics
Subjects
Details
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....bcbe7f8ebe4972300cb760adfdbae680
- Full Text :
- https://doi.org/10.48550/arxiv.2004.12149