Rigidity of group actions on solvable Lie groups
Rigidity of group actions on solvable Lie groups
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DOI：
10.1007/s002089900091
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发表时间：
2000-06
影响因子：
1.4
通讯作者：
Burkhard Wilking
中科院分区：
文献类型：
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作者：
Burkhard Wilking
我们建立类似的三个Bieberbach定理的格在一个半直积$\mathsf{K}\r\mathsf{K}$其中是一个连通的，单连通的可解李群，是它的自同构群的一个紧子群。我们首先证明了在超可解李群上的作用度量等价于在超可解李群上的作用。后者被证明是由自己的仿射同态。然后我们用代数方法把这些格刻画为多晶群。进一步地，我们将任意多晶群实现为一个格上的半直积$\mathsf{S}\r\mathsf{F}$，其中的有限群的阶由一个仅依赖于维数的常数限定.第一个比伯巴赫定理的推广也被用来得到第三个比伯巴赫定理的部分推广。最后，我们证明了对于任何挠自由闭子群$\Uppermum\subset \mathsf {K}\r times\mathsf{K}$，Uppermum是紧致流形上向量丛的全空间dB，其中B是可解李群与挠自由多晶群的商。
We establish analogs of the three Bieberbach theorems for a latticein a semidirect product $\mathsf{K}\rtimes\mathsf{K}$ whereis a connected, simply connected solvable Lie group andis a compact subgroup of its automorphism group. We first prove that the action ofonis metrically equivalent to an action ofon a supersolvable Lie group. The latter is shown to be determined byitself up to an affine diffeomorphism. Then we characterize these lattices algebraically as polycrystallographic groups. Furthermore, we realize any polycrystallographic groupas a lattice in a semidirect product $\mathsf{S}\rtimes\mathsf{F}$ withbeing a finite group whose order is bounded by a constant only depending on the dimension of. This generalization of the first Bieberbach theorem is used to obtain a partial generalization of the third one as well. Finally we show for any torsion free closed subgroup $\Upsilon \subset \mathsf{K}\rtimes\mathsf{K}$ that the quotientis the total space of a vector bundle over a compact manifoldB, whereBis the quotient of a solvable Lie group by a torsion free polycrystallographic group.