The Topological Structure of Polish Groups and Groupoids of Measure Space Transformations

Abstract

It is proved that the groupoid of nonsingular partial isomorphisms of a Lebesgue space (X, ft) is weakly contractible in a "strong" sense: we present a contraction path which preserves invariant the subgroupoid of ^-preserving partial isomorphisms as well as the group of nonsingular transformations of X. Moreover, let $ be an ergodic measured discrete equivalence relation on X. extended the weak and the uniform topology to the group Aut (X, IJL) of nonsingular transformations of (X, fJL) and generalized the results obtained in [Ha]. However, the homotopical properties of this group have not been studied so far.

Outer Automorphism Group of the Ergodic Equivalence Relation Generated by Translations of Dense Subgroup of Compact Group on its Homogeneous Space

“…(R r , m) . Moreover, this topology coincides with the Polish topology on AutRp which was considered by T. Hamachi and M. Osikawa [HO] (see [Dan,section 3]).…”

Topological properties of full groups

“…IX, Theorem 7.3]. Currently many big Polish groups are known to be homeomorphic to 2 . Examples include Aut(X, µ) with the weak topology (Nhu [24]), the group of orientation-preserving homeomorphisms of the unit interval (Anderson), the isometry group of the Urysohn space (Melleray [22]), and many others.…”

“…the set {x ∈ X : T x = x}. 2 Identifying the topological type of big symmetry groups has been an ongoing enterprise for the last few decades. During that time, infinite-dimensional topology has developed many tools which allow us to do that.…”

Connected Polish groups with ample generics

“…Full groups of type III ergodic automorphisms are contractible, as was shown by Danilenko in [Dan95]. Besides, Eigen ([Eig81]) has proved that they are simple.…”