On spectra of Koopman, groupoid and quasiregular representations
Amenability, coarse embeddability and fixed point properties December 06, 2016  December 09, 2016
Location: SLMath: Eisenbud Auditorium
Unitary representations
koopman representation
groupoid representation
spectra of representations
groups of intermediate growth
Amenability
aTmenability
fixed point properties
hyperbolic groups and generalizations
Banach space
group cohomology
expander graph
index theory
noncommutative geometry
20K40  Homological and categorical methods for abelian groups
00A35  Methodology of mathematics {For mathematics education, see 97XX}
0111  Research data for problems pertaining to history and biography
00B25  Proceedings of conferences of miscellaneous specific interest
00B55  Collections of translated articles of miscellaneous specific interest
20E15  Chains and lattices of subgroups, subnormal subgroups [See also 20F22]
14652
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 quasiregular 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
Grigorchuck Notes

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