Subperiodic groups and bounded automorphisms of periodic graphs
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866914561044512768 |
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| author | Baburin, Igor A. |
| author_facet | Baburin, Igor A. |
| contents | 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~$\R^3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_12630 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Subperiodic groups and bounded automorphisms of periodic graphs Baburin, Igor A. Group Theory Combinatorics 20-04 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~$\R^3$. |
| title | Subperiodic groups and bounded automorphisms of periodic graphs |
| topic | Group Theory Combinatorics 20-04 |
| url | https://arxiv.org/abs/2605.12630 |