Commensurability and the character variety
Commensurability and the character variety
复制标题
可通约性和性格多样性
DOI：
10.4310/mrl.1999.v6.n5.a11
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发表时间：
1999
影响因子：
1
通讯作者：
A. Reid
中科院分区：
文献类型：
--
作者：
D. Long;A. Reid
回想一下，双曲3-流形M和N被称为可重叠的，如果它们有一个公共的有限单复盖。这等价于在PSL（2，C）中具有共轭的有限指数子群的基本群。一般来说，很难确定两个流形是否可重叠，一旦最明显的可重叠不变量（例如，不变迹场，见[13]和[17]）一致。当M是具有单个尖点的有限体积双曲3-流形时，它的SL（2，C）-表示和特征变种（自始至终分别用R（M）和X（M）表示）一直是理解M的拓扑结构的基本工具，参见[6]、[5]和[4]。这些技巧可以扩展到M的PSL（2，C）-特征标簇，我们用Y（M）表示（[2]，有关详细信息，请参见§2.1）。对于SL（2）或PSL（2），我们始终使用下标0来表示X（M）（或Y（M））中包含π1（M）的忠实离散表示特征的分量。本文的主要结果是关于如何用Y 0（M）来检测不可约性。例如，主要结果之一可以总结如下（术语和定义见§2）：
Recall that hyperbolic 3-manifolds M and N are said to be commensurable if they have a common finite sheeted covering. This is equivalent to the fundamental groups having subgroups of finite index which are conjugate in PSL(2,C). In general it is very difficult to determine if two manifolds are commensurable or not, once the most obvious invariants of commensurability (for example, the invariant trace field, see [13] and [17]) agree. When M is a finite volume hyperbolic 3-manifold with a single cusp, its SL(2,C)-representation and character varieties, denoted respectively, by R(M) and X(M) throughout, have been fundamental tools in understanding the topology of M , see [6], [5], and [4]. These techniques can be extended to the PSL(2,C)-character variety of M , which we denote by Y (M) ([2], and see §2.1 for some details). Throughout, for either SL(2) or PSL(2), we use the subscript 0 to denote a component of X(M) (or Y (M)) containing the character of a faithful discrete representation of π1(M). The main results of this paper concern how Y0(M) be can used to detect incommensurability. For example, one of the main results can be summarized in the following (for terminology and definitions see §2):