Nilpotent Cantor actions

Hurder, Steven; Lukina, Olga

doi:10.1090/proc/15660

Nilpotent Cantor actions
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by Steven Hurder and Olga Lukina PDF

Proc. Amer. Math. Soc. 150 (2022), 289-304 Request permission

Abstract:

A nilpotent Cantor action is a minimal equicontinuous action $\Phi \colon \Gamma \times \mathfrak {X} \to \mathfrak {X}$ on a Cantor space $\mathfrak {X}$, where $\Gamma$ contains a finitely-generated nilpotent subgroup $\Gamma _0 \subset \Gamma$ of finite index. In this note, we show that these actions are distinguished among general Cantor actions: any effective action of a finitely generated group on a Cantor space, which is continuously orbit equivalent to a nilpotent Cantor action, must itself be a nilpotent Cantor action. As an application of this result, we obtain new invariants of nilpotent Cantor actions under continuous orbit equivalence.

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