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 \(L^0(\phi, G)\), where \(\phi\) 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 \(L^0(\phi,{\mathbb R})\) for \(\phi\) 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 \(L^0(\phi, G)\) to \(L^0(\mu, H)\), where \(\phi\) is pathological, \(\mu\) 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.
A conjecture of Hjorth states that the only countable Borel equivalence relations reducible to orbit equivalence relations of Polish abelian group actions are the hyperfinite ones. This conjecture was recently refuted by Allison, where he showed that every treeable countable Borel equivalence relation reduces to such an orbit equivalence relation. We show that this holds for more general equivalence relations, including all countable Borel equivalence relations. This is joint with Josh Frisch.