边界作用的遍历性与Nielsen-Schreier理论
Ergodic properties of boundary actions and Nielsen--Schreier theory
摘要：
研究了自由群$F$的任意子群$H$在边界$\partial F$上关于一致测度作用的基本遍历性（遍历性和保守性）.我们的方法是几何的和组合的，它是基于选择一个系统的Nielsen-Schreier生成器在$H$与测地线生成树的Schreier图$X=H\backslash F$。我们给出了几个（模0）的等价描述的保守和耗散部分的边界的Hopf分解。此外，我们与保守性和耗散性的行动与Schreier图$X$的增长和分组$H$（$\equiv$ cogrowth的$X$），分别。我们还构建了许多例子来说明各种相关概念之间的联系。
英文摘要：
We study the basic ergodic properties (ergodicity and conservativity) of the action of an arbitrary subgroup $H$ of a free group $F$ on the boundary $\partial F$ with respect to the uniform measure. Our approach is geometrical and combinatorial, and it is based on choosing a system of Nielsen--Schreier generators in $H$ associated with a geodesic spanning tree in the Schreier graph $X=H\backslash F$. We give several (mod 0) equivalent descriptions of the Hopf decomposition of the boundary into the conservative and the dissipative parts. Further we relate conservativity and dissipativity of the action with the growth of the Schreier graph $X$ and of the subgroup $H$ ($\equiv$ cogrowth of $X$), respectively. We also construct numerous examples illustrating connections between various relevant notions.
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