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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, a...
For every group $G$, of arbitrary cardinality, we construct a right-invariant metric with at most $32$ values whose isometries are exactly the permutations preserving every right-invariant metric on $G$. The proof combines subgroup-entry ranks and sign-variation colorings with a short-word rigidity theorem of Leemann a...
We prove that at every infinite cardinality, every group which is neither abelian of exponent greater than two nor generalized dicyclic admits a graphical regular representation, settling the infinite-group part of Watkins's conjecture. We also determine the Cayley index of every infinite group: it is $1$, $2$, or $8$,...
For every group $G$, of arbitrary cardinality, we construct a right-invariant metric with at most $32$ values whose isometries are exactly the permutations preserving every right-invariant metric on $G$. The proof combines subgroup-entry ranks and sign-variation colorings with a short-word rigidity theorem of Leemann a...
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