Stronger expansion for polynomial-degree abelian Cayley complexes

Determine whether polynomial-degree abelian Cayley complexes can satisfy the stronger assumptions used in low-soundness agreement testing, including coboundary expansion or cosystolic expansion of associated faces complexes.

Background

The constructions in the paper establish local spectral expansion but do not establish cofilling constants or cosystolic expansion. The paper emphasizes that applications such as low-soundness agreement testing may require stronger expansion properties than the local spectral bounds proved here.

The unresolved question is whether abelian Cayley complexes whose degree is polynomial in the ambient dimension can also meet these stronger coboundary or cosystolic expansion requirements for associated face complexes.

References

Can polynomial-degree abelian Cayley complexes satisfy the stronger assumptions used in low-soundness agreement testing, including coboundary or cosystolic expansion of associated faces complexes?

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

In order to attain good distance, we have to apply the non-Abelian lift in the construction of cubical complexes as explained in Remark \ref{remark:2D_dim} and Corollary \ref{coro:2D_dist}. However, the starting point of our method in this section assumes abelian lifts in Eq.~eq:action_coboundary_commute, and this identity does not hold for non-Abelian lifts. We leave this problem as a future research opportunity.

— Non-Abelian sheaf quantum LDPC codes: good and magical  (2609.30159 - Li et al., 24 Sep 2026) in Section 4.4, Long-range magic throughout the gauged code space