论文标题
可解决的组和仿射行动
Solvable Groups and Affine Actions on the Line
论文作者
论文摘要
我们证明了有限生成的可解决方案基团在实际间隔内（被认为是半偶联的）的结构性结果。作为应用程序，我们获得了J. F. Plante首先考虑的问题的新答案，该问题询问在哪个条件下，可解决的组在实际间隔中的作用是半偶联的，可以通过仿射转换来对该行的作用。我们表明，$ C^1 $差异性在封闭时间间隔内总是如此。对于同构的任意行动（如Plante所示），该结果不再是真实的，我们表明，在端点附近的细菌级别上，与仿射作用的半偶联性仍然存在。最终，对于包括所有可解决的线性群体在内的大量可解决的群体，我们表明，在线上的仿射作用家族是强大的，因为在线上同构的任何作用都足够接近仿射动作，必须是半偶联的。如反例所示，这种稳健性对于一般可解决的组失败了。
We prove a structural result for orientation-preserving actions of finitely generated solvable groups on real intervals, considered up to semi-conjugacy. As applications we obtain new answers to a problem first considered by J. F. Plante, which asks under which conditions an action of a solvable group on a real interval is semi-conjugate to an action on the line by affine transformations. We show that this is always the case for actions by $C^1$ diffeomorphisms on closed intervals. For arbitrary actions by homeomorphisms, for which this result is no longer true (as shown by Plante), we show that a semi-conjugacy to an affine action still exists in a local sense, at the level of germs near the endpoints. Finally for a vast class of solvable groups, including all solvable linear groups, we show that the family of affine actions on the line is robust, in the sense that any action by homeomorphisms on the line which is sufficiently close to an affine action must be semi-conjugate to an affine action. This robustness fails for general solvable groups, as illustrated by a counterexample.
