Self-replicating degrees of crystallographic groups

Abstract

An integer \(d\geq 2\) is called a self-replicating degree of a group \(G\) if there exist a subgroup \(H\leq G\) of index \(d\) and a surjective homomorphism \(\phi:H\rightarrow G\) with trivial core. These are precisely the alphabet sizes of self-replicating (recurrent) actions of \(G\). We investigate self-replicating degrees of crystallographic groups and compare them with self-covering degrees, that is, integers \(d\) for which \(G\) contains a subgroup of index \(d\) isomorphic to \(G\). Geometrically, the latter are precisely the degrees of self-coverings of the flat orbifold \(\mathbb{R}^n/G\). We establish a method for describing the self-covering and self-replicating degrees of crystallographic groups and obtain explicit polynomial descriptions for all plane and space groups.