Subperiodic groups and bounded automorphisms of periodic graphs

TL;DR

This paper classifies subperiodic groups in three dimensions, develops invariants for recognizing their isomorphism classes, and explores bounded automorphisms of Cayley graphs, with applications to embedding ladder graphs in space.

Contribution

It provides a classification of 3D subperiodic groups, introduces invariants for isomorphism recognition, and analyzes bounded automorphisms of Cayley graphs with applications to graph embeddings.

Findings

75 rod groups classified into 32 isomorphism classes

80 layer groups classified into 34 isomorphism classes

Cayley graphs of space groups have bounded automorphisms characterized by conjugation stability

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~$\R^3$.