Holomorphic polynomial crystallographic actions of nilpotent groups

Abstract

We prove that every simply connected nilpotent Lie group endowed with a left-invariant nilpotent complex structure is biholomorphic to $\mathbb{C}^n$. Moreover, we construct such a biholomorphism explicitly by polynomial maps in exponential coordinates. As a consequence, every lattice in such a Lie group admits a free, properly discontinuous and cocompact action on $\mathbb{C}^n$ by holomorphic polynomial automorphisms. We interpret this consequence as a holomorphic analogue of polynomial crystallographic actions introduced by Dekimpe, Igodt, and Lee.