$\R^{n} \rtimes G(n)$ is Algebraically Determined

Abstract

Let $G$ be a Polish (i.e., complete separable metric topological) group. Define $G$ to be an algebraically determined Polish group if for any Polish group $L$ and algebraic isomorphism $\varphi: L \mapsto G$, we have that $\varphi$ is a topological isomorphism. Let $M(n,\R)$ be the set of $n \times n$ matrices with real coefficients and let the group $G$ in the above definition be the natural semidirect product $\R^{n} \rtimes G(n)$, where $n \ge 2$ and $G(n)$ is one of the following groups: either the general linear group $GL(n,\R) = \left\{ A \in M(n,\R) \ | \ \det(A) \ne 0 \right\}$, or the special linear group $SL(n,\R) = \left\{ A \in GL(n,\R) \ | \ \det(A) = 1 \right\}$, or $|SL(n,\R)| = \left\{ A \in GL(n,\R) \ | \ |\det(A)| = 1 \right\}$ or $GL^{+}(n,\R) = \left\{ A \in GL(n,\R) \ | \ \det(A) > 0 \right\}$. These groups are of fundamental importance for linear algebra and geometry. The purpose of this paper is to prove that the natural semidirect product $\R^{n} \rtimes G(n)$ is an algebraically determined Polish group. Such a result is not true for $\complexes^{n} \rtimes GL(n,\complexes)$ nor even for $\R^{3} \rtimes SO(3,\R)$. The proof of this result is done in a sequence of steps designed to verify the hypotheses of the road map Theorem 2. A key intermediate result is that $\varphi^{-1}(SO(n,\R))$ is an analytic subgroup of $L$ for every $n \ge 2$.