Property Tau is a weaker version of Kazdan property T, where one considers only the representations which factors through a finite index subgroup. Informally, property Tau is "equivalent" to property T for the pro-finite completion of the group, but one have to keep in mind the this is far from precise definition.
In this talk I will explain that property Tau is not a pro-finite property, i.e., there exist 2 groups with the same pro-finite completion such that one has Tau and the other does not.
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