Groups without unitary representations, submeasures, and the escape property

Abstract: 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. 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 $L^0(\phi, G)$ 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.