Actions of Nilpotent Groups on Complex Algebraic Varieties

Abstract

Abstract We study nilpotent groups that act faithfully on complex algebraic varieties. In the finite case, we show that when $\textbf {k}$ is a number field, a finite $p$-subgroup of the group of polynomial automorphisms of $\textbf {k}^d$ is isomorphic to a subgroup of $\textrm {GL}_d(\textbf {k})$. In the case of infinite nilpotent group actions, we show that a finitely generated nilpotent group $H$ acting on a complex quasiprojective variety $X$ of dimension $d$ can be embedded in a $p$-adic Lie group that acts faithfully and analytically on $\textbf {Z}_p^d$. As a consequence, we show that the virtual derived length of $H$ is at most the dimension of $X$.