Compact actions whose orbit equivalence relations are not profinite

2018/07/15 by Ioana, Adrian
#Dynamical Systems (math.DS) #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.1807.05476

Abstract

Let Γ\curvearrowright (X,μ) be a measure preserving action of a countable group Γ on a standard probability space (X,μ). We prove that if the action Γ\curvearrowright X is not profinite and satisfies a certain spectral gap condition, then there does not exist a countable-to-one Borel homomorphism from its orbit equivalence relation to the orbit equivalence relation of any modular action (i.e., an inverse limit of actions on countable sets). As a consequence, we show that if Γ is a countable dense subgroup of a compact non-profinite group G such that the left translation action Γ\curvearrowright G has spectral gap, then Γ\curvearrowright G is antimodular and not orbit equivalent to any, \it not necessarily free, profinite action. This provides the first such examples of compact actions, partially answering a question of Kechris and answering a question of Tsankov.

Related

• Orbit equivalence and Borel reducibility rigidity for profinite actions with spectral gap
• Strong ergodicity, property (T), and orbit equivalence rigidity for\n translation actions
• Approximate equivalence of actions
• Orbit inequivalent actions of non-amenable groups
• From invariant measures to orbit equivalence, via locally finite groups