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More Polish full groups
We associate to every action of a Polish group on a standard probability
space a Polish group that we call the orbit full group. For discrete groups, we
recover the well-known full groups of pmp equivalence relations equipped with
the uniform topology. However, there are many new examples, such as orbit full
groups associated to measure preserving actions of locally compact groups. In
fact, we show that such full groups are complete invariants of orbit
equivalence.
We give various characterizations of the existence of a dense conjugacy class
for orbit full groups, and we show that the ergodic ones actually have a unique
Polish group topology. Furthermore, we characterize ergodic full groups of
countable pmp equivalence relations as those admitting non-trivial continuous
character representations.Comment: Comments welcome
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