Fixed words, function groups, and exact odd boundedness exponents

For an infinite zero-dimensional compact metrizable space \(X\) and a Polish group \(P\), we characterize boundedness with a fixed number of factors of \(C(X,P)\) by words alternating a target identity neighbourhood with fixed elements. If the least number of neighbourhood factors required for every identity neighbourhood of \(P\) is \(k\), then \(C(X,P)\) has least finite-neighbourhood exponent \(k\) and least open-exhaustion exponent \(2k-1\). Neither exponent is finite when no such \(k\) exists. Matching word bounds for circular covering groups then realize every odd least open-exhaustion exponent at least five, both among non-Archimedean Polish groups and among path-connected locally contractible Polish groups. We also establish an ordinary-boundedness criterion and transfer results for continuous and measurable function groups.

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