Local and end deformation theorems for uniform embeddings

A local deformation property for uniform embeddings in metric manifolds (LD) is formulated and its behavior 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 κ≤0 the κ-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 κ-cones induce contractions of some subgroups of groups of bounded uniform homeomorphisms.