Charmenability of arithmetic groups of product type

Abstract

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.

arXiv comment39 pages. v3: minor modifications. To appear in Invent. Math

References in corpus (8)

• Kesten's theorem for Invariant Random Subgroups
• Operator-algebraic superrigidity for $S{L}_n(\mathbb{Z}),n\ge 3$
• Boundaries, rigidity of representations, and Lyapunov exponents
• Stationary characters on lattices of semisimple Lie groups
• Stationary C*-dynamical systems
• Super-Rigidity and non-linearity for lattices in products
• Characters of the group ${\mathrm{EL}}_d(R)$ for a commutative Noetherian ring $R$
• On invariant random positive definite functions

Cited by in corpus (10)

• Stationary characters on lattices of semisimple Lie groups
• Noncommutative ergodic theory of higher rank lattices
• Charmenability of higher rank arithmetic groups
• Subalgebras, subgroups, and singularity
• The noncommutative factor theorem for lattices in product groups
• On relative commutants of subalgebras in group and tracial crossed product von Neumann algebras
• On the amenable subalgebras of group von Neumann algebras
• Infinite characters of type II on $S{L}_n(\mathbb{Z})$
• Full realization of ergodic IRS entropy in SL2(Z) and free groups
• Operator algebraic characterization of the noncommutative Poisson boundary