On the density of images of the power maps in Lie groups

Let G be a connected Lie group. In this paper, we study the density of the images of individual power maps (P_k:Grightarrow G:gmapsto g^k). We give criteria for the density of (P_k(G)) in terms of regular elements, as well as Cartan subgroups. In fact, we prove that if (mathrm{Reg}(G)) is the set of regular elements of G, then (P_k(G)cap mathrm{Reg}(G)) is closed in (mathrm{Reg}(G)). On the other hand, the weak exponentiality of G turns out to be equivalent to the density of all the power maps (P_k). In linear Lie groups, weak exponentiality reduces to the density of (P_2(G)). We also prove that the density of the image of (P_k) for G implies the same for any connected full rank subgroup.