Automorphism groups of dense subgroups of \(\mathbb{R}^n\) (Q1985625)

Let \(V\) be a topological vector space over the topological field \(\mathbb R\) of real numbers, \(X\) a dense subgroup of the additive group \(V(+)\) of \(V\), and \(\mathrm{HGL}(V)\) be the group of all topological automorphisms of \(V(+)\). The authors show that if \(X\) is a dense subgroup of \(V(+)\), and \(\Phi(X):=\{\alpha\in \mathrm{HGL}(V)\vert \alpha(X)=X\}\), then the mapping \(\Phi(X)\to \mathrm{Homeo}(X), \alpha\mapsto \alpha\upharpoonright_X\), where \(\alpha\upharpoonright_X\) denotes the restriction of \(\alpha\) on \(X\) and \(\mathrm{Homeo}(X)\) stands for the homeomorphism group of \(X\), is an isomorphism. Based on this result the authors raise two general problems: Problem 1. Let \(V\) be a complete topological vector space and \(X\) a dense subgroup of the additive group of \(V\). Determine the subgroup \(\Phi(X)\) of \(\mathrm{Homeo}(X)\). Problem 2. Given a complete topological vector space \(V\), which subgroups of \(\mathrm{HGL}(V)\) have the form \(\Phi(G)\) for some dense subgroup \(G\) of the additive group of \(V?\) The main results of the paper related to the Problems 1,2 are obtained for topological vector spaces \(\mathbb R,\mathbb R^n, n\in\mathbb N\), and \(\mathbb R^{\mathbb N}\). It is proved if \(G\) is a dense subgroup of \(\mathbb R^n(n\ge 1)\), then \(\mid\Phi(G)\mid\leq \mid G\mid\) (Theorem 7.2). It is constructed an example of a dense subgroup \(G\) of \(\mathbb R^{\mathbb N}\) for which \(\mid \Phi(G)\mid>\mid G\mid\). Corollary 9.6 answers partially Problem 2: Let \(n\ge 2\) and let \(H\) be a subgroup of the general linear group \(GL(n,\mathbb R)\) such that \(SO(n)\subseteq H\subsetneq GL(n,\mathbb R)\), where \(SO(n)\) is the special orthogonal group. Then \(H\neq \Phi(G)\) for any non-trivial subgroup \(G\) of \(\mathbb R^n\). In the paper are raised some open questions.