Polish full groups preserving an infinite measure
Fabien Hoareau
TL;DR
This work extends the theory of full groups beyond probability measures to σ-finite infinite measures. It proves that ergodic full groups admit at most one Polish topology refining the weak topology and coarser than the uniform topology, and that orbit full groups [r_G] are Polish under orbital convergence in measure for locally finite actions; these groups are complete invariants for orbit equivalence. It characterizes when the finitely supported subgroup G_f can be given a Polish topology, showing this occurs precisely when the full group comes from a countable equivalence relation. The paper further develops algebraic/topological properties, including simplicity of G_f, contractibility of orbit full groups, and dense/generic behavior of aperiodic/ergodic elements, culminating in a comprehensive framework for infinite-measure full groups and their orbit structures.
Abstract
We extend some results of Carderi and Le Maître 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 Maître 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.