Geometric invariants of TDLC completions
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915407268413440 |
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| author | Castellano, Ilaria Quintanilha, José Pedro |
| author_facet | Castellano, Ilaria Quintanilha, José Pedro |
| contents | Recently, Bonn and Sauer showed that, from the point of view of compactness properties, the Schlichting completion of a Hecke pair $(Γ,Λ)$ behaves precisely as if it were the quotient of $Γ$ by $Λ$. Motivated by this result, we prove that a similar phenomenon holds for the $Σ$-sets. More generally, we relate the $Σ$-sets of every TDLC completion of a Hecke pair $(Γ,Λ)$ to the $Σ$-sets of $Γ$ whenever $Λ$ satisfies suitable compactness properties. We provide applications to TDLC completions of Baumslag-Solitar groups and certain groups of upper triangular matrices studied by Schesler. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_01639 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometric invariants of TDLC completions Castellano, Ilaria Quintanilha, José Pedro Group Theory 22D05, 57M07, 20F65 Recently, Bonn and Sauer showed that, from the point of view of compactness properties, the Schlichting completion of a Hecke pair $(Γ,Λ)$ behaves precisely as if it were the quotient of $Γ$ by $Λ$. Motivated by this result, we prove that a similar phenomenon holds for the $Σ$-sets. More generally, we relate the $Σ$-sets of every TDLC completion of a Hecke pair $(Γ,Λ)$ to the $Σ$-sets of $Γ$ whenever $Λ$ satisfies suitable compactness properties. We provide applications to TDLC completions of Baumslag-Solitar groups and certain groups of upper triangular matrices studied by Schesler. |
| title | Geometric invariants of TDLC completions |
| topic | Group Theory 22D05, 57M07, 20F65 |
| url | https://arxiv.org/abs/2507.01639 |