We address the problem to characterise closed type I subgroups of the automorphism group of a tree. Even in the well-studied case of Burger-Mozes' universal groups, non-type I criteria were unknown. We prove that a huge class of groups acting properly on trees are not of type I. In the case of Burger-Mozes groups, this yields a complete classification of type I groups among them. Our key novelty is the use of von Neumann algebraic techniques to prove the stronger statement that the group von Neumann algebra of the groups under consideration is non-amenable.

中文翻译：

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作用于树的局部紧群、I 型猜想和不可修正的冯诺依曼代数

我们解决了表征树的自同构群的封闭类型 I 子群的问题。即使在经过充分研究的 Burger-Mozes 普遍群体的案例中，非 I 型标准也是未知的。我们证明在树上正确作用的一大类群不是 I 类。在 Burger-Mozes 群的情况下，这产生了其中 I 类群的完整分类。我们的主要新颖之处在于使用冯诺依曼代数技术来证明更强的陈述，即所考虑的群的群冯诺依曼代数是不适用的。