Spaces of Kleinian groups
Spaces of Kleinian groups
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DOI：
10.1007/bfb0060314
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发表时间：
1970
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通讯作者：
L. Bers
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文献类型：
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作者：
L. Bers
设G为Kleinian群。G在另一个Kleinian群上的同构，如果是在G的极限集合a上的Riemann球的拟共形自同构的共轭，则称为拟共形变形（如果a有测度o，后一个条件是空的，这可能是本文所考虑的所有群的情况）。如果G的两个拟共形变形因黎曼球的共形自同构的共轭而不同，则称为等价。我们将变形空间T (G)定义为G的拟共形变形的等价类的集合，这个定义类似于第一类Fuchsian群的Teichmliller空间T (G)的定义，但不完全相同。该空间的元素是G在另一个Fuchsian群上的拟共形变形的等价类。对于有限生成的Kleinian群G， T (G)在复数空间中存在一个自然且相当明显的嵌入，正确的问题是T (G)是否是复流形，而不是如何定义T (G)上的复结构。然而，如果G是第一类有限生成的Fuchsian群，则T (G)的自然参数化产生实数空间中的集合，正确的问题是T (G)是否允许正则复结构。众所周知，答案是肯定的，而且
Let G be a Kleinian group. An isomorphism of G onto another Kleinian group is called a quasiaonformaZ deformation if it is a conjugation by a quasiconformal automorphism of the Riemann sphere, which is conformal ae on the limit set A of G.(The latter con dition is vacuous if A has' measure O. This is likely to be the case for all groups considered in this paper.) Two quasiconformal defor mations of G are called equivaZent if they differ by a conjugation by a conformal automorphism of the Riemann sphere. We define the deformation spaae T (G) to be the set of equivalence classes of quasiconformal deformations of G. This definition is similar to, but not identical with, the definition of the Teichmliller space T (G) of a Fuchsian group of the first kind. The elements of that space are equivalence classes of quasiconformal deformations of G onto another Fuchsian group.For a finitely generated Kleinian group G there exists a natural and rather obvious embedding of T (G) into a complex number space, and the correct question to ask is whether T (G) is a complex manifold, and not how to define a complex structure on T (G). If G is a finite ly generated Fuchsian group of the first kind, however, the natural parametrization of T (G) yields a set in a real number space, and the correct question to ask is whether T (G) admits a canonical complex structure. It is well known that the answer is affirmative, and