发表机构
Harvard University（哈佛大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究自然群（群运算由度量唯一决定），利用刚性定理证明许多非交换与交换群是自然的，并提出了若干开放问题。
AI 中文摘要
一个群 (G,*) 被称为自然的，如果其群运算由 G 上的度量结构 (G,d) 唯一确定，具体含义为：对于任何群结构 (G,.)，若其所有右平移都是 (G,d) 的等距同构，则该群结构必须与 (G,*) 同构。例子包括所有连通李群或所有由对合生成的群（Sutherland）。Leemann 和 de la Salle 的定向刚性定理使得非交换结构定理得以升级：所有基数不大于连续统且非广义二面体群的非交换群都是自然的。其推论是：所有非亚贝尔李群、所有基数不大于连续统的单群、流形的所有同胚群、微分同胚群或辛同胚群、概率空间的自同构群或非交换晶体群都是自然的。交换结构定理也得到了扩展：Jarosz 的仿射刚性结合 Mazur-Ulam 定理意味着每个实或复向量空间的加法群都是自然的，且所有连通的交换巴拿赫李群都是自然的。Babai 的一个定理意味着每个布尔群对于任意基数都是自然的。尚未解决的问题包括：对于基数 k>c 和奇素数 p，C_p^k 是否为自然的；任何域 F 的加法群是否为自然的；以及非交换结构定理中的基数假设是否必要。一个主要问题是 G^2=G 是否蕴含 G 是自然的。这即使在交换群中也是开放的：2G=G 是否蕴含 G 是自然的？
英文摘要
A group (G,*) is natural if its group operation is uniquely determined by a metric structure (G,d) on G in the following sense: every group structure (G,.) for which all right translations are isometries of (G,d) must be isomorphic to (G,*). Examples included all connected Lie groups or all groups generated by involutions (Sutherland). The orientation rigidity theorem of Leemann and de la Salle allows to upgrade the non-abelian structure theorem: every non-abelian group of cardinality not larger than the continuum that is not generalized dicyclic is natural. A consequence is that all non-metabelian Lie groups, all simple group of cardinality not larger then the continuum, all homeomorphism-, diffeomorphism -or symplectomorphism groups of manifolds, or automorphism groups probability spaces or non-abelian crystallographic groups are natural. Also the abelian structure theorem is extended: affine rigidity of Jarosz, together with Mazur-Ulam's theorem implies that the additive group of every real or complex vector spaces is natural and that all connected abelian Banach Lie groups are natural. A theorem of Babai implies that every Boolean group is natural for every cardinality. Undecided is whether C_p^k is natural for cardinals k>c and odd prime p, and whether the additive group of any field F is natural, or whether the cardinality assumption in the non-abelian structure theorem is needed. A major question is whether G^2=G implies that G is natural. This is already open for abelian groups: does 2G=G imply that G is natural?
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