# American Institute of Mathematical Sciences

2017, 11: 99-123. doi: 10.3934/jmd.2017005

## On spectra of Koopman, groupoid and quasi-regular representations

 1 Department of Mathematics, University of Toronto, Room 6290, 40 St. George Street, Toronto, ON M5S 2E4, Canada 2 Department of Mathematics, MS 3368, Texas A&M University, College Station, TX 77843-3368, USA

Received  May 31, 2016 Revised  November 05, 2016 Published  January 2017

Fund Project: Both authors were supported by the Swiss National Science Foundation.
RG:Supported by NSF grant DMS-1207699 and NSA grant H98230-15-1-0328.

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.

Citation: Artem Dudko, Rostislav Grigorchuk. On spectra of Koopman, groupoid and quasi-regular representations. Journal of Modern Dynamics, 2017, 11: 99-123. doi: 10.3934/jmd.2017005
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