A group acts crystallographically on a space when the action is properly discontinuous with compact quotient; Euclidean crystallographic groups are the discrete cocompact subgroups of the isometry group, and Bieberbach's theorems make them virtually abelian and finite in number in each dimension. Enlarging the acting group to the affine group, or to the nil-affine group Aut(N) semidirect N of a simply connected nilpotent Lie group, breaks that classification and raises the Auslander conjecture and Milnor's question. Students learn how these geometric questions become algebra: left-invariant affine structures on a Lie group correspond to pre-Lie algebra structures on its Lie algebra, simply transitive nil-affine actions correspond to complete post-Lie algebra structures on a pair of Lie algebras, and commutative post-Lie (CPA) structures admit sharp classification results on perfect, complete and nilpotent Lie algebras.
A survey of crystallographic actions and of the algebraic structures that encode them. An action of a group on a space is crystallographic when it is properly discontinuous with compact quotient; a E…