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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Online Access: | https://arxiv.org/abs/2305.00610 |
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| _version_ | 1866918002631376896 |
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| author | Fraczyk, Mikolaj Lee, Minju |
| author_facet | Fraczyk, Mikolaj Lee, Minju |
| contents | Let $G$ be a connected semisimple real algebraic group and $Γ<G$ be its Zariski dense discrete subgroup. We prove that if $Γ\backslash G$ admits any finite Bowen-Margulis-Sullivan measure, then $Γ$ is virtually a product of higher rank lattices and discrete subgroups of rank one factors of $G$. This may be viewed as a measure-theoretic analogue of classification of convex cocompact actions by Kleiner-Leeb and Quint, which was conjectured by Corlette in 1994. The key ingredients in our proof are the product structure of leafwise measures and the high entropy method of Einsiedler-Katok-Lindenstrauss. In a companion paper jointly with Edwards and Oh, we use this result to show that the bottom of the $L^2$ spectrum has no atom in any infinite volume quotient of a higher rank simple algebraic group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_00610 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Discrete subgroups with finite Bowen-Margulis-Sullivan measure in higher rank Fraczyk, Mikolaj Lee, Minju Dynamical Systems Geometric Topology 37A17 Let $G$ be a connected semisimple real algebraic group and $Γ<G$ be its Zariski dense discrete subgroup. We prove that if $Γ\backslash G$ admits any finite Bowen-Margulis-Sullivan measure, then $Γ$ is virtually a product of higher rank lattices and discrete subgroups of rank one factors of $G$. This may be viewed as a measure-theoretic analogue of classification of convex cocompact actions by Kleiner-Leeb and Quint, which was conjectured by Corlette in 1994. The key ingredients in our proof are the product structure of leafwise measures and the high entropy method of Einsiedler-Katok-Lindenstrauss. In a companion paper jointly with Edwards and Oh, we use this result to show that the bottom of the $L^2$ spectrum has no atom in any infinite volume quotient of a higher rank simple algebraic group. |
| title | Discrete subgroups with finite Bowen-Margulis-Sullivan measure in higher rank |
| topic | Dynamical Systems Geometric Topology 37A17 |
| url | https://arxiv.org/abs/2305.00610 |