Short presentations for crystallographic groups

TL;DR

This paper introduces a practical method for constructing concise presentations of Euclidean crystallographic groups using generators and relations, emphasizing the link between relators and cycles in Cayley graphs.

Contribution

It provides a new approach to generate short group presentations by relating relators to cycles in Cayley graphs, with applications to high-symmetry periodic graphs.

Findings

Short presentations often correspond to strong rings in Cayley graphs.

Method applied to vertex-transitive groups acting on periodic graphs.

Relations between geodesics and cycles in quotient graphs are explored.

Abstract

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.