Borel structurability on the 2-shift of a countable group

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• 作者：Brandon Seward^a ; ^{b.m.seward@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Robin D. Tucker-Drob^b ; ^{rtuckerd@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：03E15 ; 37A35 ; 37B10 ; 22F10
• 刊名：Annals of Pure and Applied Logic
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：167
• 期：1
• 页码：1-21
• 全文大小：503 K

文摘

We show that for any infinite countable group G   and for any free Borel action G鈫稾$G鈫?/mo><mi>X</mi>$ there exists an equivariant class-bijective Borel map from X   to the free part Free(2^G)$\mathrm{Free}(2^G)$ of the 2-shift G鈫?^G$G鈫?/mo><msup><mrow><mn>2</mn></mrow><mrow><mi>G</mi></mrow></msup>$. This implies that any Borel structurability which holds for the equivalence relation generated by G鈫稦ree(2^G) 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(2^G)$\mathrm{Free}(2^G)$ 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 2^G$2^G$ lands in the free part. As a corollary we obtain that for every 系>0$系>0$, every free p.m.p. action of G has a free factor which admits a 2-piece generating partition with Shannon entropy less than 系. This generalizes a result of Danilenko and Park.