We generalise a technical tool, originally developed by Pervova for the study of maximal subgroups in Grigorchuk and GGS groups, to all weakly branch groups satisfying a natural condition, and in particular to all branch groups. We then use this tool to prove that every maximal subgroup of infinite index of a branch group is also a branch group. As a further application of this result, we show that every maximal subgroup of the Basilica group is of finite index.

中文翻译：

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关于分支群和弱分支群中无穷指数的最大子群

我们将最初由 Pervova 开发的用于研究 Grigorchuk 和 GGS 群中最大子群的技术工具推广到满足自然条件的所有弱分支群，尤其是所有分支群。然后我们用这个工具来证明一个分支群的无穷指数的每个最大子群也是一个分支群。作为这个结果的进一步应用，我们证明了 Basilica 群的每个最大子群都是有限指数的。