Bounded geometry in Polish groups: locally compact models, amenability, and simple examples

We develop locally compact models and applications for Polish groups of bounded geometry. Building on Hrushovski's quasi-model theorem and the ultrafilter framework of Krupiński–Pillay, we give an explicit coarse-density calculation for a model with compact conjugation-invariant multiplication defect and exact control of the intrinsic coarsely bounded bornology. Applications yield the Haagerup property under topological amenability and two-sided entropy estimates under metric amenability. Packing and model obstructions treat the stated classes of groups. For McCleary's classical simple periodic coloured-order group, we compute the pointwise Polish topology and intrinsic coarse geometry, obtaining a non-locally compact quasi-line.

This paper develops material from the author's monograph Property (OB) and the Coarse Geometry of Polish Groups. It is one of eight separate papers intended to replace the monograph.

GitHub repository: https://github.com/eulogica/property-ob-independent-papers