On Graphs, Groups and Geometry

Abstract

An abstract group (G,*) is natural if there exists a metric structure (G,d) on G such that (G,*) is up to abstract group isomorphisms the only group structure on G for which all right translations are isometries of (G,d). Every connected Lie group G is natural. Disconnected Lie groups can be non-natural, like R \times C_2 or Pin^-(2)=Dic(S^1,-1). There are also natural disconnected groups like O(n). Every reflection group of cardinality at most the continuum is natural. Both the infinite dihedral group D = C_2 * C_2 and modular group PSL(2,Z) = C_2 * C_3 are natural. The rational numbers Q are natural. The p-adic integers for prime p are natural if and only if p is odd. The p-adic additive groups (Q_p,+) of the fields Q_p are all natural. An arbitrary non-abelian, finitely generated group G is natural if and only if G is not generalized dicyclic. A finitely generated abelian group is natural if and only if it is finite and its Sylow 2-subgroup is elementary abelian. Examples of non-natural groups are the cyclic groups C_4m for m > 0, the quaternion group Q_8, the integers Z, or the doubled line R \times C_2. Examples of natural groups are the dihedral groups, the additive group (R^n,+), the Rubik cube, connected Lie groups and all free groups F_n for n > 1 or all Pin^(n) for n > 2, the Lorentz or Poincare group. A principal ingredient is the rigidity theory of Cayley graphs developed by Leemann and de la Salle.