Polish full groups preserving an infinite measure

TL;DR

This paper extends results on full groups from probability to infinite measure spaces, establishing uniqueness of Polish topologies, invariance under orbit equivalence, and characterizing when finitely supported elements form a Polish group.

Contribution

It generalizes key properties of full groups to the infinite measure setting, including topology uniqueness, invariance, and conditions for Polishability of finitely supported elements.

Findings

Uniqueness of Polish group topology on ergodic full groups

Orbit full groups are complete invariants for orbit equivalence

Finitely supported elements form a Polish group only for countable equivalence relations

Abstract

We extend some results of Carderi and Le Ma\^itre on full groups in the probability context to the infinite measure one: there exists at most one Polish group topology (refining the weak topology and coarser than the uniform topology) on an ergodic full group, and the orbit full group of a locally compact group acting in a Borel manner can be endowed with a Polish group topology. Moreover, orbit full groups are complete invariants for orbit equivalence. We then generalize a result from Le Ma\^itre on non-Polishability of the group of finitely supported bijections: the finitely supported elements of an ergodic full group carries a Polish group topology if and only if the associated full group comes from a countable equivalence relation. We finish with algebraic and topological results on ergodic (orbit) full groups concerning normal subgroups, contractibility and genericity of aperiodic elements.