On surjectivity of the power maps of solvable Lie groups
Let \(G\) be a connected solvable Lie group. The author finds conditions under which the power map \(P_n: G\to G\), \(P_ng=g^n\), is surjective. The most detailed results are obtained for the case where \(G\) is a semi-direct product of a compact torus and a simply connected solvable exponential group. If \(G\) is simply connected, the surjectivity of \(P_n\) (\(n\geq 2\)) is shown to be equivalent to its injectivity. In this case \(P_n\) is a diffeomorphism. Each of these properties is equivalent to the surjectivity of the exponential map.
scientific article; zbMATH DE number 1749645
scientific article