Cayley graphs with few automorphisms: the case of infinite groups. (Graphes de Cayley avec peu d’automorphismes: le cas des groupes infinis.) (English. French summary) Zbl 1536.05234
Summary: We characterize the finitely generated groups that admit a Cayley graph whose only automorphisms are the translations, confirming a conjecture by M. E. Watkins [J. Comb. Theory, Ser. B 21, 47–56 (1976; Zbl 0319.05114)]. 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.
MSC:
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
| 20B27 | Infinite automorphism groups |
| 20P05 | Probabilistic methods in group theory |
Keywords:
GRR; DRR; ORR; Cayley graph; automorphisms of graphs; generalized dihedral group; generalized dicyclic group; regular automorphism groupCitations:
Zbl 0319.05114References:
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