Near-Optimal Expanding Generating Sets for Solvable Permutation Groups

Abstract: Let $G =&lt;S&gt;$ be a solvable permutation group of the symmetric group $S_n$ given as input by the generating set $S$. We give a deterministic polynomial-time algorithm that computes an \emph{expanding generating set} of size $\tilde{O}(n<sup>2)$ for $G$. More precisely, the algorithm computes a subset $T\subset G$ of size $\tilde{O}(n<sup>2)(1/\lambda)<sup>{O(1)}$ such that the undirected Cayley graph $Cay(G,T)$ is a $\lambda$-spectral expander (the $\tilde{O}$ notation suppresses $\log <sup>{O(1)}n$ factors). As a byproduct of our proof, we get a new explicit construction of $\varepsilon$-bias spaces of size $\tilde{O}(n\poly(\log d))(\frac{1}{\varepsilon})<sup>{O(1)}$ for the groups $\Z_d<sup>n$. The earlier known size bound was $O((d+n/\varepsilon<sup>2))<sup>{11/2}$ given by \cite{AMN98}.