Intersections of topologically tame subgroups of Kleinian groups
Intersections of topologically tame subgroups of Kleinian groups
复制标题
克莱因群的拓扑驯服子群的交集
DOI：
10.1007/bf02788766
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发表时间：
1995
期刊：
影响因子：
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通讯作者：
James W. Anderson
中科院分区：
文献类型：
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作者：
James W. Anderson
本文继续研究Kleinian群Γ的两个双生成子群Γ 1和Γ 2的交，这个问题已经被许多作者研究过了.在引言的最后，我们简要介绍了这项工作。我们在这里考虑了Γ的拓扑驯服子群的交。我们主要感兴趣的是确定交Γ 1 <$Γ 2的极限集与Γ 1和Γ 2的极限集的交之间的联系;因此，我们总是假设极限集的交是非空的。这类结果的主要应用是检测一个非平凡的元素在一对子群的交集，通过寻找点在他们的极限集。我们所证明的是，模一些很好理解的例外情况下，极限集的交叉两个拓扑驯服的子群的克莱因群的极限集本质上是交叉的极限集。为了更准确地说明结果，我们看一些特殊的案例。由于两个初等子群的交是无趣的，我们将始终假设Γ 2是非初等的。一个子群Γ 1 =<$γ <$是循环分裂的情形，取决于γ是斜驶的还是抛物线的。定理A涉及γ是斜驶的情况，并指出，模一个例外情况，γ的一些幂位于Γ 2中。例外的情况是，直到有限指数子群，有限体积双曲3-流形的纤维在圆上，并具有基本群Γ，其中纤维具有基本群Γ 2，Γ由Γ 2和γ生成。定理A：设Γ是Kleinian群，Γ 2是Γ的非初等拓扑驯服子群，γ是Γ的斜驶元，固定Λ（Γ 2）的一点.然后，或者r在r 2上虚拟纤维化，或者对于某些n> 0，γ n ∈ r 2。
In this paper, we continue the investigation of intersections of pairs of finitely generated subgroups Γ1 and Γ2 of a Kleinian group Γ, a question which has been examined by a number of authors. We give a brief survey of this work at the end of the introduction. We consider here intersections of topologically tame subgroups of Γ. We are interested primarily in determining the connection between the limit set of the intersection Γ1 ∩ Γ2 and the intersection of the limit sets of Γ1 and Γ2; as such, we will always assume that the intersection of the limit sets in nonempty. The major application of this sort of result is to detect a nontrivial element in the intersection of a pair of subgroups by looking at points in the intersection of their limit sets. What we show is that, modulo some well understood exceptional cases, the limit set of the intersection of two topologically tame subgroups of a Kleinian group is essentially the intersection of the limit sets. In order to state the results more precisely, we look at particular cases. As the intersection of two elementary subgroups is uninteresting, we will always assume that Γ2 is nonelementary. The case that one subgroup Γ1 = 〈γ〉 is cyclic splits, depending on whether γ is loxodromic or parabolic. Theorem A concerns the case that γ is loxodromic, and states that, modulo one exceptional case, some power of γ lies in Γ2. The exceptional case is, up to finite index subgroups, that of a finite volume hyperbolic 3-manifold which fibers over the circle and has fundamental group Γ, where the fiber has fundamental group Γ2 and Γ is generated by Γ2 and γ. Theorem A: Let Γ be a Kleinian group, let Γ2 be a nonelementary, topologically tame subgroup of Γ, and let γ be a loxodromic element of Γ fixing a point of Λ(Γ2). Then, either Γ is virtually fibered over Γ2 or γn ∈ Γ2 for some n > 0.