Determining $\mathbb R$-Rank in Semisimple Lie Groups via uniform approximate Lattice arising as Regular Model Sets
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866908933514330112 |
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| author | Mandal, Arunava Singh, Shashank Vikram |
| author_facet | Mandal, Arunava Singh, Shashank Vikram |
| contents | Let $G$ be a linear semisimple Lie group without compact factors. We show that uniform approximate lattices $Λ$ arising as regular model sets in $G$ determine the ambient group $G$ in a strong sense. Specifically, for every non-compact Cartan subgroup $C$ of $G$, there exists $g \in G$ such that the intersection $gCg^{-1} \cap Λ^2$ is non-empty and itself forms a uniform approximate lattice, extending a classical result of Mostow for lattices. The proof relies on a Moore-type ergodicity theorem for the hull of a strong approximate lattice, proved here as a key tool. Moreover, we prove that such approximate lattices determine the $\mathbb{R}$-rank of the ambient group $G$, drawing on ideas from the work of Prasad and Raghunathan on lattices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_01970 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Determining $\mathbb R$-Rank in Semisimple Lie Groups via uniform approximate Lattice arising as Regular Model Sets Mandal, Arunava Singh, Shashank Vikram Group Theory Primary 22E40, 22E15, Secondary 20N99, 22E25 Let $G$ be a linear semisimple Lie group without compact factors. We show that uniform approximate lattices $Λ$ arising as regular model sets in $G$ determine the ambient group $G$ in a strong sense. Specifically, for every non-compact Cartan subgroup $C$ of $G$, there exists $g \in G$ such that the intersection $gCg^{-1} \cap Λ^2$ is non-empty and itself forms a uniform approximate lattice, extending a classical result of Mostow for lattices. The proof relies on a Moore-type ergodicity theorem for the hull of a strong approximate lattice, proved here as a key tool. Moreover, we prove that such approximate lattices determine the $\mathbb{R}$-rank of the ambient group $G$, drawing on ideas from the work of Prasad and Raghunathan on lattices. |
| title | Determining $\mathbb R$-Rank in Semisimple Lie Groups via uniform approximate Lattice arising as Regular Model Sets |
| topic | Group Theory Primary 22E40, 22E15, Secondary 20N99, 22E25 |
| url | https://arxiv.org/abs/2604.01970 |