The non-existence of a non-abelian nilpotent Lie group with one discrete uniform subgroup up to automorphism
Let \(G\) be a connected simply connected nilpotent Lie group and \(\mathcal{C}(G)\) the hyperspace of all closed subgroups of \(G\) equipped with the Chabauty topology; this is a compact metrizable space. An open prebase of this topology is provided by the sets \[ \mathcal{O}_1(K)= \{H\in \mathcal{C}(G):\;H\cap K = \emptyset\},\quad \mathcal{O}_2(V)= \{H\in \mathcal{C}(G):\;H\cap V \neq \emptyset\}, \] where \(V\) and \(K\) run respectively, over all open and compact subsets of \(G\). A discrete subgroup \(\Gamma\) of \(G\) is called uniform in \(G\) if \(G/\Gamma\) is compact. The set of all discrete uniform subgroups of \(G\) is denoted by \(\mathcal{S}(G)\). Let \(\mathrm{Aut}(G)\) be the group of all continuous automorphisms of \(G\). Equipped with the compact-open topology, \(\mathrm{Aut}(G)\) is a Lie group. \(\mathrm{Aut}(G)\) operates continuously on \(\mathcal{S}(G)\) (endowed with the induced topology) with operation defined by \( (\phi, \Gamma)\longmapsto\phi(\Gamma)\). The main result of the paper is the characterization of nilpotent Lie groups in which the action \(\mathrm{Aut}(G)\curvearrowright \mathcal{S}(G)\) is transitive. More precisely, the authors prove that the following statements are equivalent: {\parindent=8mm \begin{itemize}\item[(i)] the group \(G\) is abelian; \item[(ii)] the action \(\mathrm{Aut}(G)\curvearrowright \mathcal{S}(G)\) is transitive; \item[(iii)] the space \(\mathcal{S}(G)\) is connected. \end{itemize}} As an immediate consequence, the authors deduce that there is no non-abelian nilpotent Lie group with one discrete uniform subgroup up to automorphism.
scientific article; zbMATH DE number 6050560
scientific article