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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.
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Cyril Houdayer, Sven Raum, Locally compact groups acting on trees, the type I conjecture and non-amenable von Neumann algebras. Comment. Math. Helv. 94 (2019), no. 1, pp. 185–219DOI 10.4171/CMH/458