On rotarily transitive graphs

TL;DR

This paper introduces and constructs infinite rotarily transitive graphs, a class of graphs where every automorphism has a fixed point, contrasting with Cayley graphs, and explores their properties using advanced group theory and geometric methods.

Contribution

It defines rotarily transitive graphs where all automorphisms fix some vertex and constructs infinite examples using groups with specific properties and geometric configurations.

Findings

No finite rotarily transitive graph with ≥2 vertices exists.

Infinite locally finite rotarily transitive graphs are constructed.

Infinite non-locally finite transitive graphs with fixed point automorphisms are built.

Abstract

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 testifies 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 such that any non-identity element acts without fixed point. We define a class of transitive graphs which are transitive in an "absolutely non-Cayley way": we consider graphs endowed with a transitive group action such that any element of the group acts with a 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) 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 the real numbers.