Subperiodic groups and bounded automorphisms of periodic graphs

Abstract

A subperiodic group is a group of motions of $d$-dimensional Euclidean space $\R^d$ which contains a translation lattice $\Z^r$ of rank $r < d$ as a subgroup of finite index. A classification into abstract group isomorphism classes is performed for subperiodic groups in dimension~3: 75 \emph{crystallographic} rod groups ($r=1$) and 80 layer groups ($r=2$) are shown to belong to 32 and 34 isomorphism classes, respectively. An easy-to-compute set of invariants is developed for recognizing these isomorphism classes from finite presentations which makes use only of the number of subgroups up to a given finite index~$n$ ($n \leq 12$ for rod groups and $n \leq 8$ for layer groups) and how many of them are normal. Cayley graphs of rod and layer groups are used to illustrate the concept of bounded automorphisms of finite order, \emph{i.e.} those when the distance between a graph vertex and its image has an upper bound. It is proven that a Cayley graph of a crystallographic space group $G$ (in which case $r=d$) possesses bounded automorphisms of finite order, if and only if the respective inverse-closed generating set is stabilized by conjugation by an element of finite order in $G$. As an application, subperiodic groups in $\R^4$ with a three-dimensional translation lattice are used to systematically derive embeddings of three-periodic \emph{ladder graphs} in~$\R^3$.