Rigidity for $$C^1$$C1 actions on the interval arising from hyperbolicity I: solvable groups
Rigidity for $$C^1$$C1 actions on the interval arising from hyperbolicity I: solvable groups
复制标题
由双曲性 I 引起的区间上 $$C^1$$C1 动作的刚性：可解群
DOI：
10.1007/s00209-016-1790-y
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发表时间：
2013
影响因子：
0.8
通讯作者：
C. Rivas
中科院分区：
文献类型：
--
作者：
C. Bonatti;I. Monteverde;A. Navas;C. Rivas
我们考虑阿贝尔 - 循环群，其中循环因子通过双曲自同构作用在阿贝尔子群上。我们表明，如果这样一个群通过闭区间的\(C^1\)微分同胚忠实地作用，且在内部没有全局不动点，那么这个作用在拓扑上共轭于一个仿射群的作用。此外，在非阿贝尔像的情况下，我们展示了一个关于位似变换的乘数的刚性结果，尽管共轭不一定是光滑的。还提出了一些关于非可解群的推论。特别地，我们给出了新的证明/例子，说明了存在有限生成的、局部可指示的群，它们没有通过区间的\(C^1\)微分同胚的忠实作用。
We consider Abelian-by-cyclic groups for which the cyclic factor acts by hyperbolic automorphisms on the Abelian subgroup. We show that if such a group acts faithfully by $$C^1$$C1 diffeomorphisms of the closed interval with no global fixed point at the interior, then the action is topologically conjugate to that of an affine group. Moreover, in case of non-Abelian image, we show a rigidity result concerning the multipliers of the homotheties, despite the fact that the conjugacy is not necessarily smooth. Some consequences for non-solvable groups are proposed. In particular, we give new proofs/examples yielding the existence of finitely-generated, locally-indicable groups with no faithful action by $$C^1$$C1 diffeomorphisms of the interval.