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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2021
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2105.14541 |
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| _version_ | 1866916250587758592 |
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| author | Brown, Aaron Fisher, David Hurtado, Sebastian |
| author_facet | Brown, Aaron Fisher, David Hurtado, Sebastian |
| contents | We establish finiteness of low-dimensional actions of lattices in higher-rank semisimple Lie groups and establish Zimmer's conjecture for many such groups. This builds on previous work of the authors handling the case of actions by cocompact lattices and of actions by $\Sl(n,\Z)$. While the results are not sharp in all cases, they do dramatically improve all known results. The key difficulty overcome in this paper concerns escape of mass when taking limits of sequences of measures. Due to a need to control Lyapunov exponents for unbounded cocycles when taking such limits, quantitative controls on the concentration of mass at infinity are need and novel techniques are introduced to avoid ``escape of Lyapunov exponent." |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2105_14541 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Zimmer's conjecture for non-uniform lattices: escape of mass and growth of cocycles Brown, Aaron Fisher, David Hurtado, Sebastian Dynamical Systems Differential Geometry Group Theory Geometric Topology 37C85, 22E40 We establish finiteness of low-dimensional actions of lattices in higher-rank semisimple Lie groups and establish Zimmer's conjecture for many such groups. This builds on previous work of the authors handling the case of actions by cocompact lattices and of actions by $\Sl(n,\Z)$. While the results are not sharp in all cases, they do dramatically improve all known results. The key difficulty overcome in this paper concerns escape of mass when taking limits of sequences of measures. Due to a need to control Lyapunov exponents for unbounded cocycles when taking such limits, quantitative controls on the concentration of mass at infinity are need and novel techniques are introduced to avoid ``escape of Lyapunov exponent." |
| title | Zimmer's conjecture for non-uniform lattices: escape of mass and growth of cocycles |
| topic | Dynamical Systems Differential Geometry Group Theory Geometric Topology 37C85, 22E40 |
| url | https://arxiv.org/abs/2105.14541 |