We give a complete computation of the Bieri–Neumann–Strebel–Renz invariants Σm(Hn) of the Houghton groups Hn. Partial results were previously obtained by the author, with a conjecture about the full picture, which we now confirm. The proof involves covering relevant subcomplexes of an associated CAT (0) cube complex by their intersections with certain locally convex subcomplexes, and then applying a strong form of the Nerve Lemma. A consequence of the full computation is that for each 1 ≤ m ≤ n − 1, Hn admits a map onto ℤ whose kernel is of type Fm−1 but not Fm; moreover, no such kernel is ever of type Fn−1.

中文翻译：

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霍顿群的 BNSR 不变量，得出结论

我们给出了 Bieri-Neumann-Strebel-Renz 不变量 Σ 的完整计算米(Hn) 的霍顿集团Hn. 部分结果是作者之前获得的，对全貌有一个猜想，我们现在证实这一点。该证明涉及通过与某些局部凸子复合体的交集来覆盖相关 CAT (0) 立方体复合体的相关子复合体，然后应用神经引理的强形式。完整计算的结果是对于每个 1 ≤米≤n- 1,Hn允许映射到 ℤ 上，其内核是 F 类型米-1但不是 F米; 此外，没有这样的内核是 F 型的n-1.