The Poincaré and related groups are algebraically determined Polish groups

The purpose of this paper is to prove a new topological fact about the Poincar\'{e} and related groups. If $G$ is a group, say that $G$ is an algebraically determined Polish (i.e., complete separable metric topological) group if, whenever $H$ is a Polish group and $\varphi: H \to G$ is an algebraic isomorphism, then $\varphi$ is a topological isomorphism. The proper Lorentz group, the proper orthochronous Lorentz group and the Heisenberg group are examples of Polish groups that are not algebraically determined. On the other hand it will be shown that the Lorentz group, the orthochronous Lorentz group and the Poincar\'{e} group and the other closely associated semi-direct products are algebraically determined Polish groups.