On Natural Groups

Abstract

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?