2015/09/08 by Damien Gaboriau, Gaboriau, Damien, Brandon Seward +1 · 3 citations
Computer Science · Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals #Probability (math.PR) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1509.02482
openalex publication_date 2015/09/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In 1987, Ornstein and Weiss discovered that the Bernoulli 2-shift over the\nrank two free group factors onto the seemingly larger Bernoulli 4-shift. With\nthe recent creation of an entropy theory for actions of sofic groups (in\nparticular free groups), their example shows the surprising fact that entropy\ncan increase under factor maps. In order to better understand this phenomenon,\nwe study a natural generalization of the Ornstein--Weiss map for countable\ngroups. We relate the increase in entropy to the cost and to the first\n\ℓ2-Betti number of the group. More generally, we study coboundary maps\narising from simplicial actions and, under certain assumptions, relate\n\ℓ2-Betti numbers to the failure of the Juzvinski u i addition formula.\nThis work is built upon a study of entropy theory for algebraic actions. We\nprove that for actions on profinite groups via continuous group automorphisms,\ntopological sofic entropy is equal to measure sofic entropy with respect to\nHaar measure whenever the homoclinic subgroup is dense. For algebraic actions\nof residually finite groups we find sufficient conditions for the sofic entropy\nto be equal to the supremum exponential growth rate of periodic points.\n