On higher order regionally proximal relations and topological characteristic factors for group actions

TL;DR

This paper explores higher-order regionally proximal relations in group actions, introducing a new topology for algebraic analysis, and establishes connections with topological characteristic factors and recurrence properties.

Contribution

It develops an algebraic approach using a new topology to characterize higher-order proximal relations and extends recurrence results to general group actions.

Findings

Algebraic characterization of $ extbf{RP}^{[d]}$ for abelian actions.

Extension of recurrence set results to general group actions.

Identification of topological characteristic factors for cubic configurations.

Abstract

We study several aspects of higher-order regionally proximal relations for group actions. First, we develop an algebraic approach to study higher-order regionally proximal relations. To this end, we introduce a new topology on a subgroup of the universal minimal system, which can be seen as a higher-order analogue of the classical $\tau$-topology. Using this topology, we obtain an algebraic characterization of the relation $\mathbf{RP}^{[d]}$ for abelian actions. Then, we study higher-order regionally proximal relations via recurrence sets, extending results of Huang, Shao, and Ye for $\mathbb{Z}$-actions to more general group actions under suitable assumptions. We then study topological characteristic factors and prove, modulo almost one-to-one factors, that the maximal factor of order $d-1$ is the topological characteristic factor of order d for cubic configurations for arbitrary group actions, and for arithmetic progressions for finitely generated abelian group actions. As a consequence, we show that $\mathbf{RP}^{[d]}$ and $\mathbf{AP}^{[d]}$ coincide on minimal points for finitely generated abelian group actions, and we apply this to obtain results on independence along arithmetic progressions.