On spectra of Koopman, groupoid and quasi-regular representations
Location: MSRI: Simons Auditorium
spectra of representations
groups of intermediate growth
fixed point properties
hyperbolic groups and generalizations
Given an action of a countable group on a probability measure space by a measure class preserving transformations one can associate a three types of unitary representations: Koopman representation, groupoid representation, and uncountable family of quasi-regular representations defined for each orbit of the action. If additionally an element of a group algebra over the field of complex numbers is given then the corresponding operators associated with each of these representations are defined. We show that there is a strong relation between spectra of them (in the form of equality or containment). More information is known in the case when the measure is invariant or the action is Zimmer amenable (hyperfinite). The result has interpretation in the terms of weak containment of unitary representations. We illustrate the use of this result and of the corresponding techniques (based the Schreier graphs approach), and show how to compute the spectrum of the Cayley graph of the first group of intermediate growth constructed by the speaker in 1980. The talk is based on a joint paper with A.Dudko
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