On spectra of Koopman, groupoid and quasi-regular representations

TL;DR

This paper explores the spectral relationships between Koopman, groupoid, and quasi-regular representations of countable groups, revealing their equivalences under certain conditions and providing new examples of irreducible representations.

Contribution

It establishes conditions under which these representations are weakly equivalent or contained, and computes spectra for specific group actions, advancing understanding of their interrelations.

Findings

Groupoid and quasi-regular representations are weakly equivalent to Koopman representations under ergodic actions.

For hyperfinite actions, Koopman and groupoid representations are weakly equivalent.

Spectra of representations for a specific intermediate growth group are explicitly calculated.

Abstract

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.