AI 中文总结
该研究证明了欧几里得空间中与晶体根系相关的足够好的镶嵌的分段等距群是初等可和的，其后续工作将证明这类群的有限生成性和更高有限性性质。
AI 中文摘要
给定欧几里得或双曲空间的一种镶嵌，分段等距群是由将空间切割为有限多个镶嵌凸子集并将它们重新粘合在一起的元素构成的群。镶嵌的分段等距群推广了Houghton群和Thompson群V，Bieri与Sach曾对立方镶嵌的这类群进行过研究。我们证明了欧几里得空间中足够好的镶嵌（比如与晶体根系相关的镶嵌）的分段等距群的结构结果，特别是证明了它们是初等可和的；后续正在进行的工作将证明这类群的有限生成性和更高有限性性质。
英文摘要
Given a tessellation of Euclidean or hyperbolic space, the piecewise isometry group is the group whose elements are given by cutting space into finitely many tessellated convex subsets and gluing them back together. Groups of piecewise isometries of tessellations generalize Houghton's groups and Thompson's group $V$, and for cubical tessellations were studied by Bieri and Sach. We prove structure results about groups of piecewise isometries of sufficiently nice tessellations of Euclidean space, such as tessellations associated to crystallographic root systems, in particular proving that they are elementary amenable. Future work in progress will prove finite generation and higher finiteness properties.
