Let G be a totally disconnected locally compact (tdlc) group. We denote by <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\texttt{Sub}\left( G\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="monospace">Sub</mi> <mfenced close=")" open="("> <mi>G</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> the space of closed subgroups of G equipped with the Chabauty topology. A closed subgroup H of G is called locally elliptic if every compact subset of H is contained in a compact subgroup. In this paper, we address the following quention: Is the collection <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\texttt{Sub}_{\mathcal{L}\mathcal{E}}\left( G\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="monospace">Sub</mi> <mrow> <mi mathvariant="script">L</mi> <mi mathvariant="script">E</mi> </mrow> </msub> <mfenced close=")" open="("> <mi>G</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> of closed locally elliptic subgroups of G a closed subset of the Chabauty space <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\texttt{Sub}\left( G\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="monospace">Sub</mi> <mfenced close=")" open="("> <mi>G</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation>? We prove that the space of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\texttt{Sub}_{\mathcal{L}\mathcal{E}}\left( G\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="monospace">Sub</mi> <mrow> <mi mathvariant="script">L</mi> <mi mathvariant="script">E</mi> </mrow> </msub> <mfenced close=")" open="("> <mi>G</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> is Chabauty-closed when the group G contains an open solvable subgroup.

The space of closed locally elliptic subgroups of a locally compact totally disconnected group

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发表日期：

2026年7月21日

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Yousra Kammoun

摘要

Let G be a totally disconnected locally compact (tdlc) group. We denote by $\texttt{Sub}\left( G\right) $ $Sub (G)$ the space of closed subgroups of G equipped with the Chabauty topology. A closed subgroup H of G is called locally elliptic if every compact subset of H is contained in a compact subgroup. In this paper, we address the following quention: Is the collection $\texttt{Sub}_{\mathcal{L}\mathcal{E}}\left( G\right) $ ${Sub}_{L E} (G)$ of closed locally elliptic subgroups of G a closed subset of the Chabauty space $\texttt{Sub}\left( G\right) $ $Sub (G)$ ? We prove that the space of $\texttt{Sub}_{\mathcal{L}\mathcal{E}}\left( G\right) $ ${Sub}_{L E} (G)$ is Chabauty-closed when the group G contains an open solvable subgroup.