Actions of Nilpotent Groups on Complex Algebraic Varieties

We study nilpotent groups that act faithfully on complex algebraic varieties. In the finite case, we show that when k is a number field, a finite p-subgroup of the group of polynomial automorphisms of k(d) is isomorphic to a subgroup of GL(d)(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 Z(p)(d). As a consequence, we show that the virtual derived length of H is at most the dimension of X.