Efficient contractions of the Urysohn isometry group

We construct a contraction of the Urysohn isometry group with uniform and coarse control. Pointwise stabilizers of nonempty finite sets admit contractions uniformly continuous for the left uniformity, and the global contraction preserves bounded-displacement sets. Ordinary contractibility is due to Melleray, while the quantitative realization and embedding inputs are due to Lešnik. The construction here adds guided surjective back-and-forth, inverse evaluation control, and uniform endpoint estimates. A preliminary bounded-chain argument shows that the bounded-image clause of efficient contractibility follows from the stated uniform-continuity hypothesis.

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