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Abelian Cayley High-Dimensional Expanders with Polylogarithmic Degree
Abstract
We construct an explicit infinite family of simple two-dimensional Cayley complexes over 𝔽2n whose degree is polynomial in n and whose nontrivial vertex-link eigenvalues lie in [−λ,λ] for every fixed λ>0. For every fixed d≥2, we also obtain an explicit infinite family of weighted d-dimensional Cayley complexes over 𝔽2n with codimension-two local spectral norm at most 1/d and Cayley degree Θd(n). Our two-dimensional construction uses evaluation at rational points of algebraic curves to produce projective direction sets and many functions affine along these directions, which may be useful for further constructions and improvements.
Related
- Cayley graphs of abelian groups, circular colorings and discrete fundamental groups
- Fast robbers on abelian Cayley graphs and digraphs
- Isomorphisms of abelian Cayley graphs with their natural edge-colouring
- Optimal Sparsifiers for Abelian Cayley Graphs
- Exact Diameter Windows for Random Cayley Graphs on Odd-Order Abelian Groups
BibTeX
@misc{mao2026abelian,
title = {{Abelian Cayley High-Dimensional Expanders with Polylogarithmic Degree}},
author = {Songtao Mao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08937},
eprint = {2609.08937},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08937},
}
RIS
TY - UNPB
TI - Abelian Cayley High-Dimensional Expanders with Polylogarithmic Degree
AU - Songtao Mao
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08937
UR - https://arxiv.org/abs/2609.08937
AB - We construct an explicit infinite family of simple two-dimensional Cayley complexes over 𝔽_2^n whose degree is polynomial in n and whose nontrivial vertex-link eigenvalues lie in [−λ,λ] for every fixed λ>0. For every fixed d≥2, we also obtain an explicit infinite family of weighted d-dimensional Cayley complexes over 𝔽_2^n with codimension-two local spectral norm at most 1/d and Cayley degree Θ_d(n). Our two-dimensional construction uses evaluation at rational points of algebraic curves to produce projective direction sets and many functions affine along these directions, which may be useful for further constructions and improvements.
ER -