Charmenability of arithmetic groups of product type

We discuss special properties of the spaces of characters and positive definite functions, as well as their associated dynamics, for arithmetic groups of product type. Axiomatizing these properties, we define the notions of charmenability and charfiniteness and study their applications to the topolo...

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Bibliographic Details
Published in:Inventiones mathematicae 2022-09, Vol.229 (3), p.929-985
Main Authors: Bader, Uri, Boutonnet, Rémi, Houdayer, Cyril, Peterson, Jesse
Format: Article
Language:English
Subjects:
Algebra
Arithmetic
Axioms
Classification
Dynamical Systems
Functional Analysis
Group Theory
Lie groups
Mathematics
Mathematics and Statistics
Operator Algebras
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Summary:We discuss special properties of the spaces of characters and positive definite functions, as well as their associated dynamics, for arithmetic groups of product type. Axiomatizing these properties, we define the notions of charmenability and charfiniteness and study their applications to the topological dynamics, ergodic theory and unitary representation theory of the given groups. To do that, we study singularity properties of equivariant normal ucp maps between certain von Neumann algebras. We apply our discussion also to groups acting on product of trees.
ISSN:0020-9910
1432-1297
DOI:10.1007/s00222-022-01117-w