The fundamental group of a compact flat Lorentz space form is virtually polycyclic
The fundamental group of a compact flat Lorentz space form is virtually polycyclic
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紧致平面洛伦兹空间形式的基本群实际上是多环的
DOI：
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发表时间：
1983
期刊：
影响因子：
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通讯作者：
Y. Kamishima
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文献类型：
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作者：
W. Goldman;Y. Kamishima
平坦洛伦兹空间形式是测地线完备的零曲率洛伦兹流形。众所周知（参见Auslander和Markus [3]），这样的空间M可以表示为商R/Γ，其中R”是^维Minkowski空间（n等于M的维数），Γ是一组适当地不连续和自由地作用在R上的洛伦兹等距。特别地，M的泛覆盖与R等距，并且基本群ττλ（M）同构于Γ。定理设M是紧致平坦Lorentz空间形式。那么πλ（M）实际上是多循环的。回想一下，如果一个群可以通过从多个有限群和循环群的迭代扩张来构建，则它实际上是多环的。这一结果证实了Milnor [13]在特殊情况下的猜想。对于这个猜想以及另一个特殊情况的讨论，我们参考Fried & Goldman [8]。这个结果的重要性在于，它将紧致平坦洛伦兹空间形式的分类简化为关于可解李群中的李代数和格的相当基本的问题;我们希望在未来的出版物中继续进行这种分类。关于这种简化和第3维分类的描述，请参见Fried & Goldman [8];在第4维中，分类在Fried [7]中进行。Fried & Goldman [8，§1]和Kamishima [12]发展的结构理论的一个直接结果是下面的推论。设M是紧致平坦Lorentz空间形式。则M有一个有限覆盖，它是一个解流形的同构。本文的提纲如下。在第一节中，我们收集了关于^维Minkowski空间的等距群E（n -1，1）的一些基本事实。在第二节中，我们证明了定理中的特殊
A flat Lorentz space form is a geodesically complete Lorentzian manifold of zero curvature. It is well known (see Auslander & Markus [3]) that such a space M may be represented as a quotient R/Γ, where R" is an ^-dimensional Minkowski space (n equals the dimension of M) and Γ is a group of Lorentz isometries acting properly discontinuously and freely on R. In particular the universal covering of M is isometric to R and the fundamental group ττλ{M) is isomorphic to Γ. Theorem. Let M be a compact flat Lorentz space form. Then πλ{M) is virtually poly cyclic. Recall that a group is virtually polycyclic if it can be built by iterated extensions from finitely many finite groups and cyclic groups. This result affirms a conjecture of Milnor [13] in a special case. For discussion of this conjecture as well as another special case, we refer to Fried & Goldman [8]. The importance of this result is that it reduces the classification of compact flat Lorentz space forms to fairly elementary problems concerning Lie algebras and lattices in solvable Lie groups; we hope to pursue this classification in a future publication. For a description of this reduction and the classification in dimension 3, see Fried & Goldman [8]; in dimension 4 the classification is worked out in Fried [7]. One immediate consequence of the structure theory developed in Fried & Goldman [8, §1] and Kamishima [12] is the following Corollary. Let M be the compact flat Lorentz space form. Then M has a finite covering which is diffeomorphic to a solvmanifold. The outline of this paper is as follows. In the first section we collect some basic facts about the group E(n — 1,1) of isometries of ^-dimensional Minkowski space. In the second section we prove the theorem in the special