Local and end deformation theorems for uniform embeddings

TL;DR

This paper investigates deformation properties of uniform embeddings in metric manifolds, establishing conditions under which certain groups of uniform homeomorphisms can be strongly deformed, with implications for geometric group actions and end structures.

Contribution

It introduces the local deformation property (LD) and end deformation property (ED) for uniform embeddings, analyzing their behavior in various metric spaces and ends, and shows how these properties influence the structure of homeomorphism groups.

Findings

Metric manifolds with geometric group actions have property (LD).

Proper product ends with property (ED) allow strong deformation retractions of homeomorphism groups.

Alexander isotopies induce contractions in groups of bounded uniform homeomorphisms.

Abstract

A local deformation property for uniform embeddings in metric manifolds (LD) is formulated and its behaviour is studied in a formal view point. It is shown that any metric manifold with a geometric group action, typical metric spaces (Euclidean space, hyperbolic space and cylinders) and for \kappa \leq 0 the \kappa-cone ends over any compact Lipschitz metric manifolds, all of them have the property (LD). We also formulate a notion of end deformation property for uniform embeddings over proper product ends (ED). For example, the 0-cone end over a compact metric manifold has the property (ED) if it has the property (LD). It is shown that if a metric manifold M has finitely many proper product ends with the property (ED), then the group of bounded uniform homeomorphisms of M endowed with the uniform topology admits a strong deformation retraction onto the subgroup of bounded uniform homeomorphisms which are identity over those ends. We also study a role of uniform isotopies in deformation of uniform homeomorphisms and show that Alexander isotopies in \kappa-cones induce contractions of some subgroups of groups of bounded uniform homeomorphisms.