Short presentations for crystallographic groups

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 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 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 and subperiodic examples are also considered. Relations are explored between geodesics in periodic graphs and corresponding cycles in their quotients.