In this paper, we prove that the holomorphic automorphism groups of the spaces and are not isomorphic as topological groups. By making use of this fact, we establish the following characterization of the space : Let be a connected complex manifold of dimension that is holomorphically separable and admits a smooth envelope of holomorphy. Assume that the holomorphic automorphism group of is isomorphic to the holomorphic automorphism group of as topological groups. Then itself is biholomorphically equivalent to . This was first proved by us in  under the stronger assumption that is a Stein manifold.
"A group-theoretic characterization of the space obtained by omitting the coordinate hyperplanes from the complex Euclidean space, II." J. Math. Soc. Japan 58 (3) 643 - 663, July, 2006. https://doi.org/10.2969/jmsj/1156342031