Existence of a Zig-Zag–style product for bounded-degree high dimensional expanders

Determine whether there exists an algorithm, analogous to the Zig-Zag product for graphs, that given a bounded-degree high dimensional expander (HDX) as input outputs another HDX with more vertices while preserving the same bound on vertex degree and maintaining the same bound on the local spectral expansion of all links.

Background

Bounded-degree expander graphs can be grown using combinatorial products such as the Zig-Zag product, which preserve degree and expansion while increasing the number of vertices. For high dimensional expanders (HDXs), all known bounded-degree constructions rely on algebraic techniques, and no combinatorial or random bounded-degree construction is currently known.

This paper develops local lifts that preserve the degree of higher-dimensional faces and maintain local spectral expansion, but they do not preserve vertex degree. Establishing an HDX analogue of the Zig-Zag product would provide a general, algebra-free mechanism to generate larger bounded-degree HDXs while keeping their spectral properties, potentially broadening applications and understanding of HDXs.