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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.
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BibTeX
@misc{sutherland2026every,
title = {{Every Reflection Group Is Natural}},
author = {Alex J Sutherland},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05751},
eprint = {2609.05751},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05751},
}
RIS
TY - UNPB
TI - Every Reflection Group Is Natural
AU - Alex J Sutherland
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.05751
UR - https://arxiv.org/abs/2609.05751
AB - 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.
ER -