Groups without unitary representations, submeasures, and the escape property
Friedrich Martin Schneider, Sławomir Solecki
TL;DR
The paper investigates exotic topological groups—those with no nontrivial continuous unitary representations—via the L^{0} construction over pathological submeasures. It introduces the escape property as a geometric criterion that ensures homomorphism rigidity, proving that any continuous homomorphism from L^{0}(φ,G) to L^{0}(μ,H) is trivial when φ is pathological and H has escape property. The main contributions are: (i) showing that L^{0}(φ,G) is strongly exotic for any G when φ is pathological; (ii) establishing a general rigidity principle that blocks nontrivial homomorphisms into L^{0}(μ,H); (iii) deriving embedding and characterization results for pathological submeasures via related Boolean-algebra constructions D_{φ} and S(φ,G); and (iv) linking these findings to extreme amenability and whirlyness in amenable settings. Overall, the work enriches the understanding of how submeasure pathology shapes the representation theory and geometry of large topological groups, with implications for constructing exotic examples and understanding their universality.
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 $L^0(φ, G)$, 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 $L^0(φ,\mathbb{R})$ for $φ$ 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 $L^0(φ, G)$ to $L^0(μ, H)$, where $φ$ is pathological, $μ$ is a measure, $G$ is a topological group, and $H$ is a topological group with the escape property.