Finitary estimates for the distribution of lattice orbits in homogeneous spaces I: Riemannian metric
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866913491248480256 |
|---|---|
| author | Lin, Zuo Sarkar, Pratyush |
| author_facet | Lin, Zuo Sarkar, Pratyush |
| contents | Let $H < G$ both be noncompact connected semisimple real algebraic groups where the former is maximal proper and $Γ< G$ be a lattice. Building on the work of Gorodnik-Weiss, we refine their techniques and obtain effective results. More precisely, we prove effective convergence of the distribution of dense $Γ$-orbits in $G/H$ to some limiting density on $G/H$ assuming effective equidistribution of regions of maximal horospherical orbits under one-parameter diagonal flows inside a dense $H$-orbit in $Γ\backslash G$. The significance of the effectivized argument is due to the recent effective equidistribution results of Lindenstrauss-Mohammadi-Wang for $Δ(\operatorname{SL}_2(\mathbb R)) < \operatorname{SL}_2(\mathbb R) \times \operatorname{SL}_2(\mathbb R)$ and $\operatorname{SL}_2(\mathbb R) < \operatorname{SL}_2(\mathbb C)$ and arithmetic lattices $Γ$, and future generalizations in that direction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_01041 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Finitary estimates for the distribution of lattice orbits in homogeneous spaces I: Riemannian metric Lin, Zuo Sarkar, Pratyush Dynamical Systems Differential Geometry 22E40, 37A25, 37A17, 53C35 Let $H < G$ both be noncompact connected semisimple real algebraic groups where the former is maximal proper and $Γ< G$ be a lattice. Building on the work of Gorodnik-Weiss, we refine their techniques and obtain effective results. More precisely, we prove effective convergence of the distribution of dense $Γ$-orbits in $G/H$ to some limiting density on $G/H$ assuming effective equidistribution of regions of maximal horospherical orbits under one-parameter diagonal flows inside a dense $H$-orbit in $Γ\backslash G$. The significance of the effectivized argument is due to the recent effective equidistribution results of Lindenstrauss-Mohammadi-Wang for $Δ(\operatorname{SL}_2(\mathbb R)) < \operatorname{SL}_2(\mathbb R) \times \operatorname{SL}_2(\mathbb R)$ and $\operatorname{SL}_2(\mathbb R) < \operatorname{SL}_2(\mathbb C)$ and arithmetic lattices $Γ$, and future generalizations in that direction. |
| title | Finitary estimates for the distribution of lattice orbits in homogeneous spaces I: Riemannian metric |
| topic | Dynamical Systems Differential Geometry 22E40, 37A25, 37A17, 53C35 |
| url | https://arxiv.org/abs/2403.01041 |