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

TL;DR

This paper investigates when the images of power maps in connected Lie groups are dense, linking this property to regular elements, Cartan subgroups, and the group's weak exponentiality, with implications for linear Lie groups.

Contribution

It provides criteria for the density of power map images in Lie groups, relating it to regular elements, Cartan subgroups, and weak exponentiality, and extends results to full rank subgroups.

Findings

Density of power maps relates to regular elements and Cartan subgroups.

Weak exponentiality is equivalent to the density of all power maps.

Density of P_k(G) implies density in full rank subgroups.

Abstract

Let $G$ be a connected Lie group. In this paper, we study the density of the images of individual power maps $P_k:G\to G:g\mapsto g^k$. 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 ${\rm Reg}(G)$ is the set of regular elements of $G$, then $P_k(G)\cap {\rm Reg}(G)$ is closed in ${\rm Reg}(G)$. 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.