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Cayley graphs with few automorphisms: the case of infinite groups

Paul-Henry Leemann, Mikael de la Salle

Abstract

We characterize the finitely generated groups that admit a Cayley graph whose only automorphisms are the translations, confirming a conjecture by Watkins from 1976. The proof relies on random walk techniques. As a consequence, every finitely generated group admits a Cayley graph with countable automorphism group. We also treat the case of directed graphs.

Cayley graphs with few automorphisms: the case of infinite groups

Abstract

We characterize the finitely generated groups that admit a Cayley graph whose only automorphisms are the translations, confirming a conjecture by Watkins from 1976. The proof relies on random walk techniques. As a consequence, every finitely generated group admits a Cayley graph with countable automorphism group. We also treat the case of directed graphs.

Paper Structure

This paper contains 18 sections, 19 theorems, 46 equations, 1 table.

Table of Contents

  1. Introduction
  2. Consequences
  3. About the proof
  4. Acknowledgements:
  5. Group theory background
  6. Constructing Cayley graphs with few automorphisms
  7. Squares and random walks
  8. On the squares in finitely generated groups
  9. $C_G(s)$ is locally finite.
  10. Reduction to $F=\{1\}$.
  11. Reduction to ${s'}^2=1$.
  12. The group $G$ is virtually $2$-nilpotent.
  13. The group $G$ is virtually abelian.
  14. Random walks and proof of Lemma \ref{['lemma:Mikael2']}
  15. Counting triangles
  16. ...and 3 more sections

Key Result

Theorem 1.1

Every finitely generated group $G$ that is not virtually abelian admits a finite degree Cayley graph whose automorphism group is not larger than $G$ acting by left-translation.

Theorems & Definitions (39)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 29 more