Dense images of the power maps for a disconnected real algebraic group

Abstract

Let $G$ be a complex algebraic group defined over $\mathbb{R}$, which is not necessarily Zariski connected. In this article, we study the density of the images of the power maps $g\to g^k$, $k\in \mathbb{N}$, on real points of $G$, i.e., $G(\mathbb{R})$ equipped with the real topology. As a result, we extend a theorem of P. Chatterjee on surjectivity of the power map for the set of semisimple elements of $G(\mathbb{R})$. We also characterize surjectivity of the power map for a disconnected group $G(\mathbb{R})$. The results are applied in particular to describe the image of the exponential map of $G(\mathbb{R}).$

Final version to appear in J. Group Theory