Finitely generated subgroups and Chabauty topology in totally disconnected locally compact groups

Abstract

Abstract For a locally compact group 𝐺, we write S ⁢ U ⁢ B ⁢ ( G ) {\mathcal{SUB}}(G) for the space of closed subgroups of 𝐺 endowed with the Chabauty topology. For any positive integer 𝑛, we associate to 𝐺 the function δ G , n \delta_{G,n} from G n G^{n} to S ⁢ U ⁢ B ⁢ ( G ) {\mathcal{SUB}}(G) defined by δ G , n ⁢ ( g 1 , … , g n ) = gp ¯ ⁢ ( g 1 , … , g n ) , \delta_{G,n}(g_{1},\ldots,g_{n})=\overline{\mathrm{gp}}(g_{1},\ldots,g_{n}), where gp ¯ ⁢ ( g 1 , … , g n ) \overline{\mathrm{gp}}(g_{1},\ldots,g_{n}) denotes the closed subgroup topologically generated by g 1 , … , g n g_{1},\ldots,g_{n} . It would be interesting to know for which groups 𝐺 the function δ G , n \delta_{G,n} is continuous for every 𝑛. Let [ HW ] [\mathtt{HW}] be the class of such groups. Some interesting properties of the class [ HW ] [\mathtt{HW}] are established. In particular, we prove that [ HW ] [\mathtt{HW}] is properly included in the class of totally disconnected locally compact groups. The class of totally disconnected locally compact locally pronilpotent groups is included in [ HW ] [\mathtt{HW}] . Also, we give an example of a solvable totally disconnected locally compact group not contained in [ HW ] [\mathtt{HW}] .