On rotarily transitive graphs
Sébastien Martineau – GJM, Volume 10, Issue 2 (2025), 55-65.
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- décembre 30, 2025
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From the point of view of discrete geometry, the class of locally finite transitive graphs is a wide and important one. The subclass of Cayley graphs is of particular interest, as testified by the development of geometric group theory. Recall that Cayley graphs can be defined as non-empty locally finite connected graphs endowed with a transitive group action by graph-automorphism such that every non-identity element acts without fixed point.
We define a class of transitive graphs that are transitive in an “absolutely non-Cayley way”: we consider graphs endowed with a transitive group action by graph-automorphism such that every element of the group acts with at least one fixed point. We call such graphs {rotarily transitive graphs}, and we show that, even though there is no finite rotarily transitive graph with at least 2 vertices, there is an infinite, locally finite, connected rotarily transitive graph. The proof is based on groups built by Ivanov which are finitely generated, of finite exponent and have a small number of conjugacy classes.
We also build infinite transitive graphs (which are not locally finite but may be taken to be countable) any automorphism of which has a fixed point. This is done by considering “unit distance graphs” associated with the projective plane over suitable subfields of \mathbb R .
Milestones:
Received: Janvier 31, 2025
Accepted: October 03, 2025
Published: December 31, 2025
Authors:
Sébastien MartineauSorbonne University, Paris, France.