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 $X$.
We study nilpotent groups acting faithfully on complex algebraic varieties. We use a method of base change. For finite p-groups, we go from k, a number field, to a finite field in order to use counting lemmas. We show that a finite p-group of polynomial automorphisms of k d is isomorphic to a subgroup of GL d (k). For infinite groups, we go from C to Zp and use p-adic analytic tools and the theory of p-adic Lie groups. We show that a finitely generated nilpotent group H acting faithfully on a complex quasiprojective variety X of dimension d can be embedded into a p-adic Lie group acting faithfully and analytically on Z d p ; we deduce that d is larger than the virtual derived length of H.
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