Reconstruction of manifolds and subsets of normed spaces from subgroups   of their homeomorphism groups
  We prove various reconstruction theorems about open subsets of normed spaces. E.g. if the uniformly continuous homeomorphism groups of two such sets are isomorphic, then this isomorphism is induced by a uniformly continuous homeomorphism between these open sets. 
Matatyahu Rubin and Yosef Yomdin