An almost uniquely homogeneous subgroup of \({\mathbb{R}}^ n\)
A topological group whose only homeomorphisms are translations has (among other) the property that is uniquely homogeneous. A known example of \textit{J. van Mill} [Trans. Am. Math. Soc. 280, 491-498 (1983; Zbl 0573.22001)] lives in \(\ell_ 2\) where \(\ell_ 2\) is endowed with a nonstandard group structure. The authors find an example of a topological subgroup of \({\mathbb{R}}^ n\) which is close to being uniquely homogeneous as much as is possible, because a subgroup \(H\subset {\mathbb{R}}^ n\) must always have automorphisms of the form \(g(x)=-x+h\), \(h\in H\). The obtained example is a connected locally connected (n-1)-dimensional subgroup X of \({\mathbb{R}}^ n\) with the property that for every homeomorphism \(g\in H(X)\) \((= space\) of homeomorphisms of X with compact open topology) there is an \(h\in X\) such that \(g(x)=x+h\) or \(g(x)=-x+h\). Also dim H(X)\(=n-1\). This represents the main result of the paper, but there are also some applications.
topological subgroup of \({\mathbb{R}}^ n\)
scientific article; zbMATH DE number 3940282
scientific article