Property (T) and the Furstenberg Entropy of Nonsingular Actions
نویسندگان
چکیده
We establish a new characterization of property (T) in terms of the Furstenberg entropy of nonsingular actions. Given any generating measure μ on a countable group G, A. Nevo showed that a necessary condition for G to have property (T) is that the Furstenberg μ-entropy values of the ergodic, properly nonsingular G-actions are bounded away from zero. We show that this is also a sufficient condition.
منابع مشابه
Furstenberg Entropy Realizations for Virtually Free Groups and Lamplighter Groups
Let (G, μ) be a discrete group with a generating probability measure. Nevo shows that if G has property (T) then there exists an ε > 0 such that the Furstenberg entropy of any (G, μ)-stationary space is either zero or larger than ε. Virtually free groups, such as SL2(Z), do not have property (T). For these groups, we construct stationary actions with arbitrarily small, positive entropy. This co...
متن کاملWeak Equivalence of Stationary Actions and the Entropy Realization Problem
We initiate the study of weak containment and weak equivalence for μ-stationary actions for a given countable group G endowed with a generating probability measure μ. We show that Furstenberg entropy is a stable weak equivalence invariant, and furthermore is a continuous affine map on the space of stable weak equivalence classes. We prove the same for the associated stationary random subgroup (...
متن کاملAlgebra, Arithmetic and Multi-parameter Ergodic Theory
While classical ergodic theory deals largely with single ergodic transformations or flows (i.e. with actions of N,Z,R+ or R on measure spaces), many of the lattice models in statistical mechanics (such as dimer models) have multi-dimensional symmetry groups: they carry actions of Z or R with d > 1. However, the transition from Zor R-actions to multi-parameter ergodic theory presents considerabl...
متن کاملEntropy Theory without Past
This paper treats the Pinsker algebra of a dynamical system in a way which avoids the use of an ordering on the acting group. This enables us to prove some of the classical results about entropy and the Pinsker algebra in the general setup of measure preserving dynamical systems, where the acting group is a discrete countable amenable group. We prove a basic disjointness theorem which asserts t...
متن کاملMetastability and the Furstenberg-Zimmer Tower II: Polynomial and Multidimensional Szemerédi’s Theorem
The Furstenberg-Zimmer structure theorem for Z actions says that every measurepreserving system can be decomposed into a tower of primitive extensions. Furstenberg and Katznelson used this analysis to prove the multidimensional Szemerédi’s theorem, and Bergelson and Liebman further generalized to a polynomial Szemerédi’s theorem. Beleznay and Foreman showed that, in general, this tower can have...
متن کامل