Expanding groups with large diameter

TL;DR

This paper demonstrates that the spectral gap and diameter of Cayley graphs can vary significantly depending on the generating set, providing explicit constructions where one Cayley graph is an expander and another has very large diameter.

Contribution

It answers a question by Pyber and Szabó by constructing groups with bounded generating sets producing both expanders and graphs with super-polylogarithmic diameter.

Findings

Constructed groups with bounded generating sets yielding both expanders and large-diameter graphs.

Showed the dependence of spectral gap and diameter on the choice of generating set.

Reduced analysis to bounding exponential sums of permutational type.

Abstract

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 Szab\'o (2013) by exhibiting a sequence of finite groups $G_n$ with $|G_n| \to \infty$ admitting bounded generating sets $X_n,Y_n$ such that $\operatorname{Cay}(G_n,X_n)$ is an expander while $\operatorname{Cay}(G_n,Y_n)$ has super-polylogarithmic diameter. The construction uses the semidirect product $G_n = C_p^{n-1} \rtimes S_n$ with $p$ exponentially large in $n$, and the analysis reduces to bounding some exponential sums of permutational type.