Commensurability classes of arithmetic Kleinian groups and their Fuchsian subgroups
Commensurability classes of arithmetic Kleinian groups and their Fuchsian subgroups
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算术 Kleinian 群及其 Fuchsian 子群的可公度类
DOI：
10.1017/s030500410006727x
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发表时间：
1987
影响因子：
0.8
通讯作者：
A. Reid
中科院分区：
文献类型：
--
作者：
C. Maclachlan;A. Reid
算术Fuchsian群和Kleinian群都可以从四元数代数中得到（见[2,12]）。在一系列的论文（[8,9,10,11]）中，Takeuchi根据群中元素的轨迹，研究并刻画了所有有限协体积的Fuchsian群中的算术Fuchsian群。他的方法很容易适用于Kleinian群，并且我们在§3中得到了算术Kleinian群的类似表征。讨论了有限共体积Kleinian群的可公度类，证明了该算术群可以被表征为具有密集可公度子群的群。这里证明了算术Kleinian群的宽可通约性类与相应的四元数代数的同构类近似地一对一对应（定理2），并且很容易得出紧Kleinian群的宽可通约性类有无限多，因此紧双曲3流形也有无限多。
Arithmetic Fuchsian and Kleinian groups can all be obtained from quaternion algebras (see [2,12]). In a series of papers ([8,9,10,11]), Takeuchi investigated and characterized arithmetic Fuchsian groups among all Fuchsian groups of finite covolume, in terms of the traces of the elements in the group. His methods are readily adaptable to Kleinian groups, and we obtain a similar characterization of arithmetic Kleinian groups in §3. Commensurability classes of Kleinian groups of finite co-volume are discussed in [2] and it is shown there that the arithmetic groups can be characterized as those having dense commensurability subgroup. Here the wide commensurability classes of arithmetic Kleinian groups are shown to be approximately in one-to-one correspondence with the isomorphism classes of the corresponding quaternion algebras (Theorem 2) and it easily follows that there are infinitely many wide commensurability classes of cocompact Kleinian groups, and hence of compact hyperbolic 3-manifolds.