A fixed point theorem for the action of linear higher rank algebraic groups over local fields on symmetric spaces of infinite dimension and finite rank
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908968519991296 |
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| author | Viola, Federico |
| author_facet | Viola, Federico |
| contents | Let F be a non-archimedean local field of characteristic zero whose residue field has at least three elements. Let G be an almost simple linear algebraic group over F, with rank_F(G) >= 2. Let X be a simply connected symmetric space of infinite dimension and finite rank, with non-positive curvature operator. We prove that every continuous action by isometries of G on X has a fixed point. If the group G contains SL_3(F), the result holds without any assumption on the non-archimedean local field F. The result extends to cocompact lattices in G if the cardinality of the residue field of F is large enough, with a bound that depends on rank_F(G). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_05220 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A fixed point theorem for the action of linear higher rank algebraic groups over local fields on symmetric spaces of infinite dimension and finite rank Viola, Federico Group Theory Metric Geometry Representation Theory Let F be a non-archimedean local field of characteristic zero whose residue field has at least three elements. Let G be an almost simple linear algebraic group over F, with rank_F(G) >= 2. Let X be a simply connected symmetric space of infinite dimension and finite rank, with non-positive curvature operator. We prove that every continuous action by isometries of G on X has a fixed point. If the group G contains SL_3(F), the result holds without any assumption on the non-archimedean local field F. The result extends to cocompact lattices in G if the cardinality of the residue field of F is large enough, with a bound that depends on rank_F(G). |
| title | A fixed point theorem for the action of linear higher rank algebraic groups over local fields on symmetric spaces of infinite dimension and finite rank |
| topic | Group Theory Metric Geometry Representation Theory |
| url | https://arxiv.org/abs/2505.05220 |