Logic Seminar

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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 L0(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 L0(ϕ,R) for ϕ pathological.
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 L0(ϕ,G) to L0(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.
This is joint work with F. Martin Schneider.