Holomorphic polynomial crystallographic actions of nilpotent groups

Shuho Kanda

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arXiv · 2606.04996

Holomorphic polynomial crystallographic actions of nilpotent groups

Shuho Kanda

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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.

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Shuho Kanda. 2026-06-03. Holomorphic polynomial crystallographic actions of nilpotent groups. https://arxiv.org/abs/2606.04996

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