Flexibility of Group Actions on the Circle
Flexibility of Group Actions on the Circle
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圆上团体行动的灵活性
作者：
Sang;T. Koberda;Mahan Mj
在这个部分说明性的专著中，我们开发了一个一般框架，用于产生某些经典研究群作用于圆的奇异作用的不可数族。我们证明了如果$L$是一个非平凡极限群，则非线性表示变量$ mathm {homo}(L, mathm {homo}_+(S^1))$包含了$S^1$上忠实作用的无数个半共轭类，它们具有成对不相交的旋转谱（除了$0$），使得每个表示都提升到$mathbb{R}$。对于大多数Fuchsian群$L$，我们进一步证明了这种柔性现象甚至在局部发生，从而补充了K. Mann的结果。我们证明了$S^1$上的每一个非初等自由群或曲面群都存在一个作用，该作用与$ mathm {Homeo}_+(S^1)$上的有限维连通李子群的任何因子作用都不是半共轭的。证明了有界曲面的映射类群在$S^1$上具有非半共轭忠实作用。在建立这些结果的过程中，我们证明了$ mathm {PSL}_2(mathbb{R})$不连续子群的一般组合定理，该定理适用于大多数Fuchsian群和所有极限群。我们还给出了拓扑Baumslag引理，以及表示成Baire拓扑群的一般组合定理。作为推论，这些群上的$mathbb{Z}$-值次加性缺陷- 1拟同态的丰度。我们还通过证明它们都是等价的，给出了现存文献中各种半共轭概念的基本自包含的调和。
In this partly expository monograph we develop a general framework for producing uncountable families of exotic actions of certain classically studied groups acting on the circle. We show that if $L$ is a nontrivial limit group then the nonlinear representation variety $mathrm{Hom}(L,mathrm{Homeo}_+(S^1))$ contains uncountably many semi-conjugacy classes of faithful actions on $S^1$ with pairwise disjoint rotation spectra (except for $0$) such that each representation lifts to $mathbb{R}$. For the case of most Fuchsian groups $L$, we prove further that this flexibility phenomenon occurs even locally, thus complementing a result of K. Mann. We prove that each non-elementary free or surface group admits an action on $S^1$ that is never semi-conjugate to any action that factors through a finite--dimensional connected Lie subgroup in $mathrm{Homeo}_+(S^1)$. It is exhibited that the mapping class groups of bounded surfaces have non-semi-conjugate faithful actions on $S^1$. In the process of establishing these results, we prove general combination theorems for indiscrete subgroups of $mathrm{PSL}_2(mathbb{R})$ which apply to most Fuchsian groups and to all limit groups. We also show a Topological Baumslag Lemma, and general combination theorems for representations into Baire topological groups. The abundance of $mathbb{Z}$--valued subadditive defect--one quasimorphisms on these groups would follow as a corollary. We also give a mostly self-contained reconciliation of the various notions of semi-conjugacy in the extant literature by showing that they are all equivalent.
伪阿诺索夫流的自由 Seifert 片段
影响因子：
2
作者：
Barbot, Thierry;Fenley, Sérgio R
通讯作者：
Fenley, Sérgio R