Short presentations for crystallographic groups
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866916919235313664 |
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| author | Baburin, Igor A. |
| author_facet | Baburin, Igor A. |
| contents | A practical approach is proposed to construct short presentations for Euclidean crystallographic groups in terms of generators and relations. For our purposes a short presentation is the one with a small number of short relators for a given generating set. The connection is emphasized between relators of a group presentation and cycles in the associated Cayley graph. It is shown by examples that a short presentation is usually the one where relators correspond to strong rings in the Cayley graph and therefore provide a natural upper bound for their size. Presentations are computed for vertex-transitive groups which act with trivial vertex stabilizers on a number of high-symmetry 2-, 3- and 4-periodic graphs. Higher-dimensional as well as subperiodic examples are also considered. Relations are explored between geodesics in periodic graphs and corresponding cycles in their quotients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_19171 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Short presentations for crystallographic groups Baburin, Igor A. Group Theory Combinatorics 20-04 A practical approach is proposed to construct short presentations for Euclidean crystallographic groups in terms of generators and relations. For our purposes a short presentation is the one with a small number of short relators for a given generating set. The connection is emphasized between relators of a group presentation and cycles in the associated Cayley graph. It is shown by examples that a short presentation is usually the one where relators correspond to strong rings in the Cayley graph and therefore provide a natural upper bound for their size. Presentations are computed for vertex-transitive groups which act with trivial vertex stabilizers on a number of high-symmetry 2-, 3- and 4-periodic graphs. Higher-dimensional as well as subperiodic examples are also considered. Relations are explored between geodesics in periodic graphs and corresponding cycles in their quotients. |
| title | Short presentations for crystallographic groups |
| topic | Group Theory Combinatorics 20-04 |
| url | https://arxiv.org/abs/2508.19171 |