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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.11388 |
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Table of 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.