Groups without unitary representations, submeasures, and the escape property

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 \(L^0(\phi , G)\) $L^0(\varphi ,G)$ , where \(\phi \) $\varphi$ 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 \(L^0(\phi ,{{\,\mathrm{\mathbb {R}}\,}})\) $L^0(\varphi ,\mathbb{R})$ for \(\phi \) $\varphi$ 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 \(L^0(\phi , G)\) $L^0(\varphi ,G)$ to \(L^0(\mu , H)\) $L^0(\mu ,H)$ , where \(\phi \) $\varphi$ is pathological, \(\mu \) $\mu$ is a measure, G is a topological group, and H is a topological group with the escape property.