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.

v1: 17 pages v2: 19 pages, improvements in the presentation, Section 6 added. To appear in Annales Henri Lebesgue

References in corpus (3)

• Classification of finite groups that admit an oriented regular representation
• Commuting probabilities of infinite groups
• Cayley graphs with few automorphisms

Cited by in corpus (2)

• Measure equivalence rigidity of $\mathrm{Out}(F_N)$
• Most rigid representation and Cayley index of finitely generated groups