Generating Geometric Graphs Using Automorphisms

We outline an algorithm for generating nets, a “net” being a geometric graph embedded in a space on which some group acts transitively. We prove that any connected net in such a space can be generated by this process, which can also be regarded as a covering of the net by copies of a connected “transversal” subnet. This paper outlines some of the underlying theory supporting a crystal prediction and design program currently under development. The paper begins with some background in geometric group theory.