Polynomial-degree abelian Cayley complexes below the 1/d threshold

Establish whether, for every fixed dimension d≥3 and every 0<λ<1/d, there exist explicit families of abelian Cayley complexes over 𝔽₂ⁿ with codimension-two local spectral norm at most λ and Cayley degree polynomial in n.

Background

The paper constructs weighted d-dimensional abelian Cayley complexes with Cayley degree Θ_d(n) and codimension-two local spectral norm at most 1/d. This achieves linear degree at the endpoint 1/d, but does not address whether polynomial degree is possible for strictly stronger local spectral expansion, namely for any λ<1/d when d≥3.

The unresolved issue concerns extending the higher-dimensional construction beyond the spectral threshold attained by the graph-product method, while retaining explicitness, abelian Cayley symmetry, and polynomial degree in the ambient dimension n.

References

It remains open whether, for every fixed $d\ge3$ and $0<\lambda<1/d$, there are explicit families of abelian Cayley complexes over $_2n$ with codimension-two local spectral norm at most $\lambda$ and Cayley degree polynomial in $n$.

— Abelian Cayley High-Dimensional Expanders with Polylogarithmic Degree  (2609.08937 - Mao, 8 Sep 2026) in Section 1, subsection “Our results”

For fixed $0<\lambda<1/2$, can the two-dimensional degree in Theorem~\ref{thm:main-two-dimensional} be reduced to $n{1+o(1)}$, or $O_\lambda(n)$?

— Abelian Cayley High-Dimensional Expanders with Polylogarithmic Degree  (2609.08937 - Mao, 8 Sep 2026) in Section 6, “Discussion and open problems,” second item