Pointwise bounded cocycles and coarse decompositions of topological groups

We prove a uniform-boundedness principle for Borel cocycles between Polish groups: pointwise coarsely bounded fibres are uniformly coarsely bounded over each coarsely bounded subset of the acting group. A bounded-defect version applies to quasicocycles. The principle yields a Zappa–Szép coarse-product criterion without local boundedness and a nonsplit-extension criterion using Borel sections and pointwise intrinsic bounds, without a centralizing-section hypothesis. We also prove that local boundedness lifts through ultralocally bounded quotients. Supplementary sections give conditional coupling-composition results and examine category assumptions in factorization, using a classical additive-graph construction.

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