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Expanding groups with large diameter

Sean Eberhard, Luca Sabatini

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 Szabó (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.

Expanding groups with large diameter

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 Szabó (2013) by exhibiting a sequence of finite groups with admitting bounded generating sets such that is an expander while has super-polylogarithmic diameter. The construction uses the semidirect product with exponentially large in , and the analysis reduces to bounding some exponential sums of permutational type.
Paper Structure (6 sections, 10 theorems, 30 equations)

This paper contains 6 sections, 10 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Expansion in semidirect products
  4. Kazhdan constant and semidirect products
  5. Reduction to the exponential sum estimate
  6. Permutational exponential sums

Key Result

Theorem 1.2

There exist constants $\delta, \varepsilon > 0$ and a sequence of finite groups $(G_n)_{n \geq 1}$ with $|G_n| \to \infty$ with bounded-size generating sets $X_n, Y_n \subseteq G_n$ such that

Theorems & Definitions (20)

  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 10 more