AI 中文总结
通过粗等价和拟等距对可数Stone空间的同胚群进行分类，证明三类群分别对应粗等价类，其中第二类群拟等距于Hamming立方体。
AI 中文摘要
为了发展非局部紧拓扑群的几何群论工具，我们给出了这类群族在粗等价下的首批完整分类之一，并在可能的情况下给出了拟等距分类。在之前的一篇论文中，我们将可数Stone空间的同胚群分为三类：粗有界、无界但由粗有界集生成、无界且不由任何粗有界集生成。现在我们证明这些就是粗等价类：同一类中的任意两个群实际上是粗等价的。此外，我们证明第二类群拟等距于Hamming立方体，即由具有有限多个非零项的无穷二进制序列组成的空间，配备Hamming距离。作为证明的一部分，我们证明有限字母表上的无穷Hamming图都是双Lipschitz等价的。
英文摘要
Towards developing the tools of geometric group theory for non-locally compact topological groups, we give one of the first complete classifications of a family of such groups up to coarse equivalence, and when possible, up to quasi-isometry. In a previous paper, we placed the homeomorphism groups of countable Stone spaces into three classes: coarsely bounded, unbounded yet generated by a coarsely bounded set, and unbounded but not generated by any coarsely bounded set. Now we show that these are the coarse equivalence classes: Any two groups within one of these classes are in fact coarsely equivalent. Furthermore, we show that groups in the second class are quasi-isometric to the Hamming cube, the space comprising infinite binary sequences with finitely many nonzero entries equipped with the Hamming distance, and that groups in the third class are coarsely equivalent to the set of leaves of the regular one-ended tree of countably infinite valence. As part of the proof, we show that infinite Hamming graphs over finite alphabets are all bi-Lipschitz equivalent and prove a coarse geometric classification result for topological groups admitting exhaustions by proper, open, coarsely bounded subgroups.
Commentsv2: 20 pages, 4 figures, improvements to section 5; submitted version
