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Groups that produce expander graphs

Luca Sabatini

TL;DR

The paper addresses when sequences of finite groups or transitive group actions admit boundedly generated expanders in Cayley and Schreier graphs. It develops a framework tying expansion to isoperimetric numbers, abelianization growth $ab_k(G)$, and representation growth $Rep_k(G)$, and shows that expansions place strong constraints on these invariants. A central contribution is proving that infinite amenable groups and solvable groups of bounded derived length do not produce expander Schreier graphs, generalizing Lubotzky–Weiss from Cayley to Schreier settings, and demonstrating that poor expansion properties of a group action cannot in general be detected solely by abelian sections or by representations above a stabilizer. The results are complemented by concrete examples, such as $\mathrm{Aff}_1(p)\curvearrowright \mathbb{F}_p$, and by open problems that connect abelianization and representation growth, inviting further study of expanding versus non-expanding group sequences, including anabelian cases.

Abstract

We survey the known group properties that a sequence of finite groups or group actions needs to satisfy to admit subsets of bounded cardinality producing expander Cayley or Schreier graphs. We prove that an infinite amenable group and solvable groups of bounded derived length do not produce expander Schreier graphs, generalizing with easier proofs results of Lubotzky and Weiss for Cayley graphs. In particular, the poor expansion properties of a group action cannot in general be detected by looking at the abelian sections or at the representations above the stabilizer of a point.

Groups that produce expander graphs

TL;DR

The paper addresses when sequences of finite groups or transitive group actions admit boundedly generated expanders in Cayley and Schreier graphs. It develops a framework tying expansion to isoperimetric numbers, abelianization growth , and representation growth , and shows that expansions place strong constraints on these invariants. A central contribution is proving that infinite amenable groups and solvable groups of bounded derived length do not produce expander Schreier graphs, generalizing Lubotzky–Weiss from Cayley to Schreier settings, and demonstrating that poor expansion properties of a group action cannot in general be detected solely by abelian sections or by representations above a stabilizer. The results are complemented by concrete examples, such as , and by open problems that connect abelianization and representation growth, inviting further study of expanding versus non-expanding group sequences, including anabelian cases.

Abstract

We survey the known group properties that a sequence of finite groups or group actions needs to satisfy to admit subsets of bounded cardinality producing expander Cayley or Schreier graphs. We prove that an infinite amenable group and solvable groups of bounded derived length do not produce expander Schreier graphs, generalizing with easier proofs results of Lubotzky and Weiss for Cayley graphs. In particular, the poor expansion properties of a group action cannot in general be detected by looking at the abelian sections or at the representations above the stabilizer of a point.

Paper Structure

This paper contains 10 sections, 12 theorems, 29 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Expanding sequences of groups and actions
  3. Isoperimetric number, expander graphs
  4. Abelian sections
  5. Representation growth
  6. Amenable actions and Theorems \ref{['thAmenableSG']} and \ref{['thMain']}
  7. Amenable groups and actions
  8. Fø lner condition on the disjoint union
  9. Solvable finite groups
  10. Open problems

Key Result

Theorem 1.3

Let $G$ be an infinite amenable group acting on the finite sets $(\Omega_n)_{n \geq 1}$. Fix a finite symmetric set $S \subseteq G$. Then $(\mathop{\mathrm{Sch}}(G \circlearrowright \Omega_n,S))_{n \geq 1}$ is not a sequence of expanders.

Figures (2)

  • Figure 1: $\mathop{\mathrm{Sch}}( \, \mathop{\mathrm{Aff}}_1(13) \circlearrowright \mathbb{F}_{13} \, , \, \{ \pm 1, \cdot 2^{\pm 1} \} \, )$.
  • Figure 2: The relations between the group properties BAb, expanding, and the conditions in LW93 and HRV93.

Theorems & Definitions (27)

  • Definition
  • Definition
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Definition
  • Remark 2.1
  • Theorem 2.2: Th. 3.6 in LW93
  • Lemma 2.3
  • ...and 17 more