Watkins's conjecture holds for all infinite groups

Abstract

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 $8$, according to its algebraic type, and in every case the index is attained by a connected Cayley graph. For every infinite group G of cardinality κ, we construct 2^κ pairwise nonisomorphic Cayley graphs with exactly the unavoidable inverse-pair symmetries, diameter two, and κ common neighbors at every distinct pair. The principal tool recovers a continuous ordinal hierarchy from an alternating adjacency baseline with bounded-degree errors: robust finite patterns identify the initial classes, successive twin quotients recover the layers, and their finite exception packets determine the translation action. The reconstruction applies without a group action and is stable under additional layerwise bounded-degree edits. Further results give closed Cantor-cube families with prescribed finite data in the regular cases, sharp cofinality-dependent graph properties, and optimal three-valued shortest-path metrics.

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BibTeX RIS

BibTeX

@misc{sutherland2026watkinss,
  title = {{Watkins's conjecture holds for all infinite groups}},
  author = {Alex J. Sutherland},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.01049},
  eprint = {2610.01049},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.01049},
}

RIS

TY  - UNPB
TI  - Watkins's conjecture holds for all infinite groups
AU  - Alex J. Sutherland
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2610.01049
UR  - https://arxiv.org/abs/2610.01049
AB  - 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 $8$, according to its algebraic type, and in every case the index is attained by a connected Cayley graph. For every infinite group G of cardinality κ, we construct 2^κ pairwise nonisomorphic Cayley graphs with exactly the unavoidable inverse-pair symmetries, diameter two, and κ common neighbors at every distinct pair. The principal tool recovers a continuous ordinal hierarchy from an alternating adjacency baseline with bounded-degree errors: robust finite patterns identify the initial classes, successive twin quotients recover the layers, and their finite exception packets determine the translation action. The reconstruction applies without a group action and is stable under additional layerwise bounded-degree edits. Further results give closed Cantor-cube families with prescribed finite data in the regular cases, sharp cofinality-dependent graph properties, and optimal three-valued shortest-path metrics.
ER  -

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