The continuity of convex cores with respect to the geometric topology
The continuity of convex cores with respect to the geometric topology
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DOI：
10.4310/cag.2005.v13.n3.a1
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发表时间：
2005
影响因子：
0.7
通讯作者：
Ken'ichi Ohshika
中科院分区：
文献类型：
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作者：
Ken'ichi Ohshika
凸核的体积、拉普拉斯谱的底部和极限集的豪斯多夫维数是双曲3-流形/克莱因群的重要不变量，它们彼此密切相关。这些不变量关于克莱因群收敛的行为是一个值得研究的有趣问题，而且，预计这些不变量对于研究变形空间的拓扑结构将很有用。 Canary、Taylor、McMullen 等人研究了这些不变量在克莱因群的代数和几何收敛方面的行为。 （参见[10]，[27]，[31]。）为了研究这个问题，有必要看看相应的双曲3-流形的凸核是否在格罗莫夫意义上几何上收敛到极限的凸核。考虑一系列 Kleinian 群 {Gi} 几何收敛于 G∞。克莱因群 G 的极限集 ΛG 与 S2 ∞ 和 G 的 Nielsen 凸包在 H3 ∪ S2 ∞ 中的闭包的交点一致。如果凸核 C(H/Gi) 几何收敛于 C(H/G∞)，则就 H3 的豪斯多夫拓扑而言，Gi 的 Nielsen 凸包收敛于 G∞ 的 Nielsen 凸包。这意味着极限集 ΛGi 相对于 S2 ∞ 上的 Hausdorff 拓扑收敛到 ΛG∞。相反，正如鲍迪奇[4]所示，如果极限集ΛGi收敛到ΛG∞，那么凸核C(H/Gi)在格罗莫夫意义上几何收敛到C(H/G∞)。我们对以下问题感兴趣。 H/Gi 的凸核与 H/G∞ 的凸核几何收敛吗？或者等价地，关于 Hausdorff 拓扑，极限集 ΛGi 是否收敛到 ΛG∞？
The volume of the convex core, the bottom of the spectrum of the Laplacian, and the Hausdorff dimension of the limit set are important invariants of hyperbolic 3-manifolds/Kleinian groups, which are closely related to one another. The behaviour of such invariants with respect to the convergence of Kleinian groups is an interesting problem to study, and also, it is expected that these invariants would be useful to investigate the topological structure of deformation spaces. The behaviour of these invariants with respect to algebraic and geometric convergence of Kleinian groups has been studied by Canary, Taylor, McMullen, among others. (See [10], [27], [31].) For studying this problem, it is essential to see whether convex cores of the corresponding hyperbolic 3-manifolds converge to the convex core of the limit geometrically in the sense of Gromov. Consider a sequence of Kleinian groups {Gi} converging geometrically to G∞. The limit set ΛG of a Kleinian group G coincides with the intersection of S2 ∞ and the closure of the Nielsen convex hull of G in H3 ∪ S2 ∞. If the convex cores C(H/Gi) converge geometrically to C(H/G∞), then the Nielsen convex hulls of Gi converge to that of G∞ with respect to the Hausdorff topology of H3. This implies that the limit sets ΛGi converge to ΛG∞ with respect to the Hausdorff topology on S2 ∞. Conversely, as was shown by Bowditch [4], if the limit sets ΛGi converge to ΛG∞ , then the convex cores C(H/Gi) converge to C(H/G∞) geometrically in the sense of Gromov. We are interested in the following question. Do the convex cores of H/Gi converge geometrically to that of H/G∞? Or equivalently, do the limit sets ΛGi converge to ΛG∞ with respect to the Hausdorff topology?