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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2402.11388 |
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| _version_ | 1866929736809185280 |
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| author | Schneider, Friedrich Martin Solecki, Sławomir |
| author_facet | Schneider, Friedrich Martin Solecki, Sławomir |
| contents | We give new examples of topological groups that do not have non-trivial continuous unitary representations, the so-called exotic groups. We prove that all groups of the form $L^0(ϕ, G)$, where $ϕ$ is a pathological submeasure and $G$ is a topological group, are exotic. This result extends, with a different proof, a theorem of Herer and Christensen on exoticness of $L^0(ϕ,\mathbb{R})$ for $ϕ$ pathological. It follows that every topological group embeds into an exotic one. In our arguments, we introduce the escape property, a geometric condition on a topological group, inspired by the solution to Hilbert's fifth problem and satisfied by all locally compact groups, all non-archimedean groups, and all Banach--Lie groups. Our key result involving the escape property asserts triviality of all continuous homomorphisms from $L^0(ϕ, G)$ to $L^0(μ, H)$, where $ϕ$ is pathological, $μ$ is a measure, $G$ is a topological group, and $H$ is a topological group with the escape property. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_11388 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Groups without unitary representations, submeasures, and the escape property Schneider, Friedrich Martin Solecki, Sławomir Representation Theory Functional Analysis Group Theory 22A10, 22A25, 28A60, 28B10 We give new examples of topological groups that do not have non-trivial continuous unitary representations, the so-called exotic groups. We prove that all groups of the form $L^0(ϕ, G)$, where $ϕ$ is a pathological submeasure and $G$ is a topological group, are exotic. This result extends, with a different proof, a theorem of Herer and Christensen on exoticness of $L^0(ϕ,\mathbb{R})$ for $ϕ$ pathological. It follows that every topological group embeds into an exotic one. In our arguments, we introduce the escape property, a geometric condition on a topological group, inspired by the solution to Hilbert's fifth problem and satisfied by all locally compact groups, all non-archimedean groups, and all Banach--Lie groups. Our key result involving the escape property asserts triviality of all continuous homomorphisms from $L^0(ϕ, G)$ to $L^0(μ, H)$, where $ϕ$ is pathological, $μ$ is a measure, $G$ is a topological group, and $H$ is a topological group with the escape property. |
| title | Groups without unitary representations, submeasures, and the escape property |
| topic | Representation Theory Functional Analysis Group Theory 22A10, 22A25, 28A60, 28B10 |
| url | https://arxiv.org/abs/2402.11388 |