Cayley graphs with few automorphisms: the case of infinite groups

TL;DR

This paper characterizes finitely generated groups with Cayley graphs that have minimal automorphism groups, confirming a longstanding conjecture and showing every such group can have a Cayley graph with countable automorphisms.

Contribution

It proves a conjecture by Watkins from 1976, identifying groups with Cayley graphs only automorphic under translations, and extends results to directed graphs.

Findings

Finitely generated groups with minimal automorphism Cayley graphs are characterized.

Every finitely generated group admits a Cayley graph with countable automorphism group.

The proof uses random walk techniques.

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.