设$BS(1,n) =< a, b \ | \ aba^{-1} = b^n >$是可解Baumslag-Solitar群，其中$ n\geq 2$。已知BS（1，n）同构于由实直线的两个仿射映射$f_0(x) = x + 1$和$h_0(x) = nx $生成的群。本文讨论了BS（1，n）在闭可定向曲面上作用的动力学性质.我们在$\TT ^2$上证明了一个没有有限轨道的光滑BS（1，n）作用，研究了它及其$C^1$ -扰动的动力学行为，证明了它不是局部刚性的.本文对闭曲面$S$上的忠实拓扑BS（1，n）-作用进行了一般动力学研究。我们证明了这样的行动$<f,h \ | \ h \circ f \circ h^{-1} = f^n>$承认一个极小集包括在$fix(f)$，$f$的不动点集，只要$fix(f)$是不空的。当$S= \TT^2$时，证明了存在正整数$N$，使得$fix(f^N)$非空且包含动作的极小集.作为推论，我们得到BS（1，n）在$\TT^2$上不存在极小忠实拓扑作用.当曲面$S$的亏格至少为2，是闭的且可定向的，且$f$对恒等式是同位素的，则$fix(f)$非空且包含作用的极小集.此外，如果动作是$C^1$，则$fix(f)$包含任何极小集。
英文摘要：
Let $BS(1,n) =< a, b \ | \ aba^{-1} = b^n >$ be the solvable Baumslag-Solitar group, where $ n\geq 2$. It is known that BS(1,n) is isomorphic to the group generated by the two affine maps of the real line: $f_0(x) = x + 1$ and $h_0(x) = nx $.
This paper deals with the dynamics of actions of BS(1,n) on closed orientable surfaces. We exhibit a smooth BS(1,n) action without finite orbits on $\TT ^2$, we study the dynamical behavior of it and of its $C^1$-pertubations and we prove that it is not locally rigid.
We develop a general dynamical study for faithful topological BS(1,n)-actions on closed surfaces $S$. We prove that such actions $<f,h \ | \ h \circ f \circ h^{-1} = f^n>$ admit a minimal set included in $fix(f)$, the set of fixed points of $f$, provided that $fix(f)$ is not empty.
When $S= \TT^2$, we show that there exists a positive integer $N$, such that $fix(f^N)$ is non-empty and contains a minimal set of the action. As a corollary, we get that there are no minimal faithful topological actions of BS(1,n) on $\TT^2$.
When the surface $S$ has genus at least 2, is closed and orientable, and $f$ is isotopic to identity, then $fix(f)$ is non empty and contains a minimal set of the action. Moreover if the action is $C^1$ then $fix(f)$ contains any minimal set.
设$BS(1,n)= \langle a,b : a b a ^{-1} = b ^n\rangle$是$n \geq 2$的可解Baumslag-Solitar群。我们研究了$BS(1, n)$在平面上的保向同胚表示，假设$a$是线性映射，$b$是有界位移映射.我们发现，忠实行动的可能性在很大程度上取决于若尔当典范形的地图$h$定义的行动$a$。如果$h$在$\mathbb R$上可对角化，我们将给出例子或证明依赖于特征值的刚性定理。我们还表明，在$h$是椭圆或抛物线的情况下，一些刚性。然后利用环面的同胚给出了$BS(1, n)$作用的应用.
Actions of solvable Baumslag-Solitar groups on surfaces with (pseudo-)Anosov elements
设$BS(1,n)= <a,b : a b a ^{-1} = b ^n>$是可解Baumslag-Solitar群，其中$n \geq 2$。本文研究了用（伪）Anosov元的闭曲面的同胚表示$BS(1, n)$。也就是说，我们考虑一个封闭的表面$S$，和同胚$f, h: S \to S$，使$h f h^{-1} = f^n$，对于某些$ n\geq 2$。已知$f$（或$f$的某个幂）必须与恒等式同伦。假设$h$是具有拉伸因子$λ>1$的伪Anosov。我们证明了如果$λ> n$，则$<f,h>$不是$BS(1, n)$的忠实表示.此外，我们表明，有没有忠实的表示$BS(1, n)$的环面同胚与$h$ Anosov映射和$f$面积保持（无论$λ$的值）。
Boundary actions of Bass-Serre Trees and the applications to $C^*$-algebras
Institute for Advanced Study in Mathematics, Harbin Institute of Technology（哈尔滨工业大学数学高等研究院）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）；Beijing International Center for Mathematical Research, Peking University（北京大学北京国际数学研究中心）
