Cost, ℓ2-Betti
numbers and the sofic entropy of some algebraic actions
D. Gaboriau, B. Seward
Journal d'Analyse
Mathématique, 139 (2019), 1-65.
arXiv: http://arxiv.org/abs/1509.02482v2
HAL: https://hal.archives-ouvertes.fr/ensl-01195906v2
Download: pdf (694K)
Abstract
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. With the recent creation of an entropy theory for actions
of sofic groups (in particular free groups), their example shows the
surprising fact that entropy can increase under factor maps. In
order to better understand this phenomenon, we study a natural
generalization of the Ornstein--Weiss map for countable groups. We
relate the increase in entropy to the cost and to the first ℓ2-Betti number of the group.
More generally, we study coboundary maps arising from simplicial
actions and, under certain assumptions, relate ℓ2-Betti numbers to the failure
of the Juzvinskii addition formula. This work is built upon a study
of entropy theory for algebraic actions. We prove that for actions
on profinite groups via continuous group automorphisms, topological
sofic entropy is equal to measure sofic entropy with respect to Haar
measure whenever the homoclinic subgroup is dense. For algebraic
actions of residually finite groups we find sufficient conditions
for the sofic entropy to be equal to the supremum exponential growth
rate of periodic points.
Keyword: Sofic entropy, algebraic actions, periodic points,
Juzvinskiĭ addition formula, ℓ2-Betti
numbers, cost, Rokhlin entropy, Ornstein-Weiss map.
2010 Mathematics Subject Classification (MSC2010):
Primary 37A15, 37A35, 37B40; Secondary 37A20, 37B10