Motivated by the Bruhat and Cartan decompositions of general linear groups over local fields, we enumerate double cosets of the group of label-preserving automorphisms of a label-regular tree over the fixator of an end of the tree and over maximal compact open subgroups. This enumeration is used to show that every continuous homomorphism from the automorphism group of a label-regular tree has closed range.

中文翻译：

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自同态树群的分解定理

受局部域上一般线性群的 Bruhat 和 Cartan 分解的启发，我们在树末端的固定器和最大紧开子群上枚举了标签正则树的标签保留自同构群的双陪集。此枚举用于表明标签正则树的自同构群中的每个连续同态具有封闭范围。