Expanding groups with large diameter

We study how the spectral gap and diameter of Cayley graphs depend strongly on the choice of generating set. We answer a question of Pyber and Szabó (2013) by exhibiting a sequence of finite groups $G_n$ with $\mathrm{\mid }{G}_n\mathrm{\mid }\to \mathrm{\infty }$ admitting bounded generating sets $X_n,Y_n$ such that $\mathrm{Cay}(G_n,X_n)$ is an expander while $\mathrm{Cay}(G_n,Y_n)$ has super-polylogarithmic diameter. The construction uses the semidirect product $G_n=C_p^{n-1}⋊S_n$ with $p$ exponentially large in $n$, and the analysis reduces to bounding some exponential sums of permutational type.