作用于树的群的指示性、剩余有限性和单子项
Indicability, residual finiteness, and simple subquotients of groups acting on trees
摘要：
我们建立了三个独立的结果，对树木的群体。第一个是指一个紧生成的局部紧群，它连续作用在一个局部有限树上，具有幂零局部作用，且没有全局不动点，它是虚可指示的;也就是说，它有一个有限指数子群，它覆盖在$\mathbf{Z}$上。第二类保证了非离散局部紧群的乘积中的不可约余紧格，使得其中一个因子作用在具有幂零局部作用的树上是点传递的，不能是剩余有限的。这是来自一般的结果，独立的利益，不可约格的产品组。第三个是指具有任意给定局部作用的树的每一个非离散Burger-Mozes泛自同构自同构群都存在一个具有非离散单商的紧生成闭子群。作为应用，我们回答了D.通过证明两个正则树的乘积中某个格的非剩余有限性，得到了C. Reid关于局部紧群的结构理论。
英文摘要：
We establish three independent results on groups acting on trees. The first implies that a compactly generated locally compact group which acts continuously on a locally finite tree with nilpotent local action and no global fixed point is virtually indicable; that is to say, it has a finite index subgroup which surjects onto $\mathbf{Z}$. The second ensures that irreducible cocompact lattices in a product of non-discrete locally compact groups such that one of the factors acts vertex-transitively on a tree with a nilpotent local action cannot be residually finite. This is derived from a general result, of independent interest, on irreducible lattices in product groups. The third implies that every non-discrete Burger-Mozes universal group of automorphisms of a tree with an arbitrary prescribed local action admits a compactly generated closed subgroup with a non-discrete simple quotient. As applications, we answer a question of D. Wise by proving the non-residual finiteness of a certain lattice in a product of two regular trees, and we obtain a negative answer to a question of C. Reid, concerning the structure theory of locally compact groups.
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