Cayley graphs with few automorphisms: the case of infinite groups
Annales Henri Lebesgue, Volume 5 (2022), pp. 73-92.

Keywords GRR, DRR, ORR, Cayley graph, automorphisms of graphs, generalized dihedral group, generalized dicyclic group, regular automorphism group

### 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.

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