2016
What is it about?
A reasonable (nonempty, connected and locally finite) graph is a Cayley graph if and only if it admits a group action that is transitive on the vertices and such that any nontrivial element of the group has no fixed point. In this work, we study graphs that are transitive due to rotations, i.e. graphs that admit a group action that is transitive on the vertices and such that any element of the group has a fixed point. It is easy to show that no there is no nontrivial finite graph that is transitive under rotations. By using a Tarski monster built by Ivanov, we show that there exist infinite reasonable such graphs. Besides, by working in the framework of metric geometry à la Beckman–Quarles, we construct an example of a non locally-finite graph that is rotarily transitive in a strong sense: its full group of automorphisms acts by rotations.