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Entropy, Shannon orbit equivalence, and sparse connectivity
We say that two free p.m.p. actions of countable groups are Shannon orbit equivalent if there is an orbit equivalence between them whose associated...
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Ergodic theorems for the shift action and pointwise versions of the Abért-Weiss theorem
Let Γ be a countably infinite group. A common theme in ergodic theory is to start with a probability measure-preserving (p.m.p.) action Γ ↷ ( X, μ )...
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The Systems with Almost Banach-Mean Equicontinuity for Abelian Group Actions
In this paper, we present the concept of Banach-mean equicontinuity and prove that the Banach-, Weyl- and Besicovitch-mean equicontinuities of a...
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Kazhdan groups have cost 1
We prove that every countably infinite group with Kazhdan’s property (T) has cost 1, answering a well-known question of Gaboriau. It remains open if...
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Distributed algorithms, the Lovász Local Lemma, and descriptive combinatorics
In this paper we consider coloring problems on graphs and other combinatorial structures on standard Borel spaces. Our goal is to obtain sufficient...
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The Proximal Relation, Regionally Proximal Relation and Banach Proximal Relation for Amenable Group Actions
In this paper, we study the proximal relation, regionally proximal relation and Banach proximal relation of a topological dynamical system for...
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Cost, ℓ2-Betti numbers and the sofic entropy of some algebraic actions
In 1987, Ornstein and Weiss discovered that the Bernoulli 2-shift over the rank two free group factors onto the seemingly larger Bernoulli 4-shift....
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Full Groups
The notion of full group was introduced in 1959 by H. Dye in his study of orbit equivalence of measured dynamical systems [19] and [20]. In the first...
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Følner tilings for actions of amenable groups
We show that every probability-measure-preserving action of a countable amenable group G can be tiled, modulo a null set, using finitely many finite...
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Lück’s Approximation Theorem
After motivating and presenting the statement of Lück’s approximation theorem, we prepare the proof with a detailed synopsis of spectral calculus for...
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Invariant random subgroups of linear groups
Let Γ < GL n ( F ) be a countable non-amenable linear group with a simple, center free Zariski closure. Let Sub(Γ) denote the space of all subgroups of...
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Stable actions and central extensions
A probability-measure-preserving action of a countable group is called stable if its transformation-groupoid absorbs the ergodic hyperfinite...
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Entropy for Actions of Sofic Groups
In Section 9.3 we arrived at a definition of entropy for p.m.p. actions of amenable groups by...
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Orbit Equivalence Beyond Amenability
The Ornstein-Weiss theorem (Theorem 4.84 ) asserts that the ergodic p.m.p. actions of countably...