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Logic Seminar

Wednesday, April 17, 2024
12:00pm to 1:00pm
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Groups without unitary representations, submeasures, and the escape property
Sławomir Solecki, Department of Mathematics, Cornell University,

Please note that the time is PST

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.

For more information, please contact Math Dept. by phone at 626-395-4335 or by email at [email protected].