Every Reflection Group Is Natural

Abstract

By a reflection group we mean an abstract group generated by nonidentity involutions. Knill proved that every reflection group of cardinality at most the continuum is natural, in the sense that some metric determines its group structure up to isomorphism among group structures whose right translations are isometries, and asked whether the cardinality hypothesis can be removed. We answer this question affirmatively. More precisely, every reflection group G admits a right invariant metric taking at most seven values for which the full isometry group is the right-regular action of G. The proof replaces distinct numerical labels on generators by a finite-valued encoding of a rigid graph on a well-ordered irredundant generating set.