Chabauty limits of open and almost connected subgroups of a locally compact group

Abstract Let 𝐺 be a locally compact group. The Chabauty space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">S</m:mi> <m:mo>⁢</m:mo> <m:mi mathvariant="script">U</m:mi> <m:mo>⁢</m:mo> <m:mi mathvariant="script">B</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \mathcal{SUB}(G) , consisting of the closed subgroups of 𝐺 endowed with the Chabauty topology, provides a compact topological framework for understanding convergence phenomena among subgroups. In this paper, we investigate the Chabauty limits of two distinguished families. In the first part, we characterize locally compact groups that arise as Chabauty limits of their closed almost connected subgroups, showing that this occurs precisely when the quotient <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>G</m:mi> <m:mo>/</m:mo> <m:msub> <m:mi>G</m:mi> <m:mn>0</m:mn> </m:msub> </m:mrow> </m:math> G/G_{0} is compactly ruled. In the second part, we study the Chabauty closure of the set of open subgroups. For pro-Lie groups, we show that this closure coincides with the set of closed subgroups containing the identity component. We also provide a new characterization of totally disconnected locally compact groups in terms of the Chabauty limits of their open subgroups.