Abelian Cayley High-Dimensional Expanders with Polylogarithmic Degree

We construct an explicit infinite family of simple two-dimensional Cayley complexes over ${\mathbb{F}}_2^n$ whose degree is polynomial in $n$ and whose nontrivial vertex-link eigenvalues lie in $[-\lambda ,\lambda ]$ for every fixed $\lambda >0$. For every fixed $d\ge 2$, we also obtain an explicit infinite family of weighted $d$-dimensional Cayley complexes over ${\mathbb{F}}_2^n$ with codimension-two local spectral norm at most $1\mathrm{/}d$ and Cayley degree ${\mathrm{\Theta }}_d(n)$. 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.