论文标题
在图，组和几何形状上
On Graphs, Groups and Geometry
论文作者
论文摘要
如果（x，d）确定集合x上的同构独特的组结构（x，+），则表示公制空间（x，d）是自然的，使所有组翻译和组反转是异构体。如果这样的天然度量出现，则称为天然。如果（x，d）带有大地测量d的（x，d）自然是自然的，则为自然。我们在这里查看一些示例和一些有限组的图形规则表示始终是自然图，或者是有限图的香农产物的直接产品始终保留了自然的特性。有限天然群的半主导产物也是自然的，因为它们由合适的Cayley图的锯齿形产品代表。因此，花圈产品保留了自然群体。例如，魔方很自然。有限生成的天然群体的免费产品也是自然的。一个主要的主题是，通过将其扩展到Coxeter群体，通常可以升级非天然群体以成为自然。非天然组的示例是循环基团，其顺序由4，四个组，整数，灯塔组，自由组或P-ADIC整数组排除。原型特征是扩展整数并获取无限的二面体组，用两个自由反射代替单个发电机。最后，我们简短地讨论了将二面体群作为动态系统理论的物理时期的假设。
A metric space (X,d) is declared to be natural if (X,d) determines an up to isomorphism unique group structure (X,+) on the set X such that all the group translations and group inversion are isometries. A group is called natural if it emerges like this from a natural metric. A simple graph X is declared to be natural if (X,d) with geodesic metric d is natural. We look here at some examples and some general statements like that the graphical regular representations of a finite group is always a natural graphs or that the direct product on groups or the Shannon product of finite graphs preserves the property of being natural. The semi-direct product of finite natural groups is natural too as they are represented by Zig-Zag products of suitable Cayley graphs. It follows that wreath products preserve natural groups. The Rubik cube for example is natural. Also free products of finitely generated natural groups are natural. A major theme is that non-natural groups often can be upgraded to become natural by extending them to become Coxeter groups. Examples of non-natural groups are cyclic groups whose order is divisible by 4, the quaternion group, the integers, the lamplighter group, the free groups or the group of p-adic integers. The prototype feature is to extend the integers and get the infinite dihedral group, replacing the single generator by two free reflections. We conclude with a short discussion of the hypothesis of using the dihedral group as a physical time in dynamical system theory.
