论文标题
吉赛克的手术双曲线理想单纯歧管
Surgeries of the Gieseking hyperbolic ideal simplex manifold
论文作者
论文摘要
在我们的Novi Sad会议论文（1999年）中，我们描述了著名的Gieseking（1912）双曲线理想单纯歧管$ \ MATHCAL {S} $，导致紧凑的基本域$ \ Mathcal {s}}（s}}（k）（k）$，$ k = 2、3，\ d contern \ dots \ dots \ dots \ dots unt of con $ k $ k $ k $ k $ k $ k $ k $ k $ k $ k. $ 2（k-1）/k $。我们还计算了$ s（k）$的体积，如果$ k $输入无限，则倾向于零。那个时候，我们天真地认为我们获得了上述令人惊讶的特性的Orbifolds。作为数学评论者。 Rev.，Kevin P. Scannell（MR1770996（2001G：57030））正确地说：“这与D. A. Kazhdan和G. A. Margulis（1968年）的著名定理相抵触，并与Thurston的工作描述了大型Dehn填充物的几何相反。在本文中，我们刷新了以前的出版物。正确地，我们获得了Cone歧管（$ k> 2 $），正如A. D. Mednykh和V. S. Petrov（2006）友好地指出的那样。我们完成了讨论，并以两种几何形式得出上述锥歧管系列（GIES.1和GIES.2），通过任何理想的单纯形成对称性。此外，我们获得了第二个Orbifold系列（GIES.3和4），趋向于常规的理想单纯形，作为原始的Gieseking歧管。
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.
