Exotic proper actions on homogeneous spaces via convex cocompact representations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908391749713920 |
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| author | Bochenski, Maciej Morita, Yosuke |
| author_facet | Bochenski, Maciej Morita, Yosuke |
| contents | We construct a series of homogeneous spaces G/H of reductive type which admit proper actions of discrete subgroups of G isomorphic to cocompact lattices of O(n,1) (n=2,3,4) but do not admit proper actions of non-compact semisimple subgroups of G. The existence of such homogeneous spaces was previously not known even for n=2. Our construction of proper actions of discrete subgroups is based on Guéritaud-Kassel's work on convex cocompact subgroups of O(n,1) and Danciger-Guéritaud-Kassel's work on right-angled Coxeter groups. On the other hand, the non-existence of proper actions of non-compact semisimple subgroups is proved by the theory of nilpotent orbits and elementary combinatorics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_14274 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Exotic proper actions on homogeneous spaces via convex cocompact representations Bochenski, Maciej Morita, Yosuke Group Theory Geometric Topology Representation Theory Primary 57S30, Secondary 17B08, 20H10, 22E40, 22F30 We construct a series of homogeneous spaces G/H of reductive type which admit proper actions of discrete subgroups of G isomorphic to cocompact lattices of O(n,1) (n=2,3,4) but do not admit proper actions of non-compact semisimple subgroups of G. The existence of such homogeneous spaces was previously not known even for n=2. Our construction of proper actions of discrete subgroups is based on Guéritaud-Kassel's work on convex cocompact subgroups of O(n,1) and Danciger-Guéritaud-Kassel's work on right-angled Coxeter groups. On the other hand, the non-existence of proper actions of non-compact semisimple subgroups is proved by the theory of nilpotent orbits and elementary combinatorics. |
| title | Exotic proper actions on homogeneous spaces via convex cocompact representations |
| topic | Group Theory Geometric Topology Representation Theory Primary 57S30, Secondary 17B08, 20H10, 22E40, 22F30 |
| url | https://arxiv.org/abs/2501.14274 |