论文标题
具有山雀独立性（P）的树木上作用的小组
Groups acting on trees with Tits' independence property (P)
论文作者
论文摘要
1970年的J.山雀上有关作用在树木上的小组的文章引入了独立性属性$（\ mathrm {p}）$，作为产生非线性非差异局部紧凑型简单组的第一个例子的条件，回答了J. P. Serre的问题。在最近的完全断开的本地紧凑（T.D.L.C.）组的理论的最新发展中，该特性变得非常重要，而紧凑的简单T.D.L.C.的大部分新结构中的大部分都是新的结构。使用$（\ mathrm {p}）$或相关想法的组。 在本文中，我们旨在通过开发对古典贝司理论的“局部行动”补充来推动作用在树木上的群体的全球理论。我们描述了一个封闭的集团$ g $的$ g $的汽车（不一定是本地有限的）树$ t $ t $的东西，称为本地动作图：以$ g $的本地操作装饰的图表。局部动作图在我们的理论中起作用，类似于低音 - 塞雷理论中的群体图。代替组图的通用覆盖，我们定义了局部动作图的通用组。在这种情况下，组$ \ mathbf {u}（f）$和$ \ mathbf {u}（f_1，f_2）$扮演与hnn扩展的类似角色，分别在贝斯 - serre理论中分别与bass-serre理论合并免费产品。然后，我们展示如何直接从局部动作图直接确定通用组是否具有某些特性，例如几何密度，紧凑的产生和简单性。 我们的理论使我们能够彻底描述所有封闭的树木的自动形态群体，它们具有山雀的独立性$（\ mathrm {p}）$：它们正是本地动作图的通用组。
A 1970 article of J. Tits concerning groups acting on trees introduced an independence property $(\mathrm{P})$ as a condition to produce the first examples of nonlinear nondiscrete locally compact simple groups, answering a question of J. P. Serre. This property has become very important in the recent development of the theory of totally disconnected, locally compact (t.d.l.c.) groups, with the majority of new constructions of compactly generated simple t.d.l.c. groups using $(\mathrm{P})$ or related ideas. In this paper we aim to advance the local-to-global theory of groups acting on trees by developing a `local action' complement to classical Bass--Serre theory. We describe, for a closed group $G$ of automorphisms of a (not necessarily locally finite) tree $T$ something called a local action diagram: a graph decorated with the local actions of $G$. A local action diagram plays a role in our theory that is analogous to a graph of groups in Bass--Serre theory. In place of the universal cover of a graph of groups, we define the universal group of a local action diagram. In this context, the groups $\mathbf{U}(F)$ and $\mathbf{U}(F_1, F_2)$ play analogous roles to the HNN extension and amalgamated free product respectively in Bass--Serre theory. We then show how to determine whether the universal group has certain properties, such as geometric density, compact generation and simplicity, directly from the local action diagram. Our theory allows us to completely describe all closed groups of automorphisms of trees with Tits' independence property $(\mathrm{P})$: they are precisely the universal groups of local action diagrams.
