TY - JOUR

T1 - Higher order regionally proximal equivalence relations for general minimal group actions

AU - Glasner, Eli

AU - Gutman, Yonatan

AU - Ye, Xiang Dong

N1 - Publisher Copyright:
© 2018

PY - 2018/7/31

Y1 - 2018/7/31

N2 - We introduce higher order regionally proximal relations suitable for an arbitrary acting group. For minimal abelian group actions, these relations coincide with the ones introduced by Host, Kra and Maass. Our main result is that these relations are equivalence relations whenever the action is minimal. This was known for abelian actions by a result of Shao and Ye. We also show that these relations lift through extensions between minimal systems. Answering a question by Tao, given a minimal system, we prove that the regionally proximal equivalence relation of order d corresponds to the maximal dynamical Antolín Camarena–Szegedy nilspace factor of order at most d. In particular the regionally proximal equivalence relation of order one corresponds to the maximal abelian group factor. Finally by using a result of Gutman, Manners and Varjú under some restrictions on the acting group, it follows that the regionally proximal equivalence relation of order d corresponds to the maximal pronilfactor of order at most d (a factor which is an inverse limit of nilsystems of order at most d).

AB - We introduce higher order regionally proximal relations suitable for an arbitrary acting group. For minimal abelian group actions, these relations coincide with the ones introduced by Host, Kra and Maass. Our main result is that these relations are equivalence relations whenever the action is minimal. This was known for abelian actions by a result of Shao and Ye. We also show that these relations lift through extensions between minimal systems. Answering a question by Tao, given a minimal system, we prove that the regionally proximal equivalence relation of order d corresponds to the maximal dynamical Antolín Camarena–Szegedy nilspace factor of order at most d. In particular the regionally proximal equivalence relation of order one corresponds to the maximal abelian group factor. Finally by using a result of Gutman, Manners and Varjú under some restrictions on the acting group, it follows that the regionally proximal equivalence relation of order d corresponds to the maximal pronilfactor of order at most d (a factor which is an inverse limit of nilsystems of order at most d).

KW - Maximal pronilfactor

KW - Minimal t.d.s

KW - Regionally proximal

UR - http://www.scopus.com/inward/record.url?scp=85048298806&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2018.05.023

DO - 10.1016/j.aim.2018.05.023

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AN - SCOPUS:85048298806

SN - 0001-8708

VL - 333

SP - 1004

EP - 1041

JO - Advances in Mathematics

JF - Advances in Mathematics

ER -