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On spectra of Koopman, groupoid and quasi-regular representations

  • Author Bio: Artem Dudko artem.dudko@utoronto.ca; Rostislav Grigorchuk grigorch@math.tamu.edu
Both authors were supported by the Swiss National Science Foundation.
RG:Supported by NSF grant DMS-1207699 and NSA grant H98230-15-1-0328.
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  • In this paper we investigate relations between Koopman, groupoid and quasi-regular representations of countable groups. We show that for an ergodic measure class preserving action of a countable group G on a standard Borel space the associated groupoid and quasi-regular representations are weakly equivalent and weakly contained in the Koopman representation. Moreover, if the action is hyperfinite then the Koopman representation is weakly equivalent to the groupoid. As a corollary of our results we obtain a continuum of pairwise disjoint pairwise equivalent irreducible representations of weakly branch groups. As an illustration we calculate spectra of regular, Koopman and groupoid representations associated to the action of the 2-group of intermediate growth constructed by the second author in 1980.

    Mathematics Subject Classification: Primary: 20C15, 37A15.


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  • Figure 1.  The set $\Omega$

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