Abstract
The actions of certain nonamenable groups on the Lebesgue space are studied. An example is constructed of a group which has a continuum of weakly nonequivalent actions of type II 1. It is also proved that, if a free group with two generators has hyperfinite action on the Lebesgue space, then at least one generator acts dissipatively. A hyperfinite action is constructed for any nonamenable group.
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