Borel structurability on the 2-shift of a countable group
Academic Article
-
- Overview
-
- Research
-
- Identity
-
- Additional Document Info
-
- View All
-
Overview
abstract
-
© 2015 Elsevier B.V.. We show that for any infinite countable group G and for any free Borel action G↷X there exists an equivariant class-bijective Borel map from X to the free part Free(2G) of the 2-shift G↷2G. This implies that any Borel structurability which holds for the equivalence relation generated by G↷Free(2G) must hold a fortiori for all equivalence relations coming from free Borel actions of G. A related consequence is that the Borel chromatic number of Free(2G) is the maximum among Borel chromatic numbers of free actions of G. This answers a question of Marks. Our construction is flexible and, using an appropriate notion of genericity, we are able to show that in fact the generic G-equivariant map to 2G lands in the free part. As a corollary we obtain that for every ε>0, every free p.m.p. action of G has a free factor which admits a 2-piece generating partition with Shannon entropy less than ε. This generalizes a result of Danilenko and Park.
published proceedings
-
Annals of Pure and Applied Logic
author list (cited authors)
-
Seward, B., & Tucker-Drob, R. D.
citation count
complete list of authors
-
Seward, Brandon||Tucker-Drob, Robin D
publication date
publisher
published in
Research
keywords
-
Bernoulli Shift
-
Borel Combinatorics
-
Borel Reducibility
-
Borel Structurability
-
Entropy
-
Factor Map
Identity
Digital Object Identifier (DOI)
Additional Document Info
start page
end page
volume
issue