Limit group invariants for non-free Cantor actions

Author:

HURDER STEVENORCID,LUKINA OLGA

Abstract

A Cantor action is a minimal equicontinuous action of a countably generated group $G$ on a Cantor space $X$. Such actions are also called generalized odometers in the literature. In this work, we introduce two new conjugacy invariants for Cantor actions, the stabilizer limit group and the centralizer limit group. An action is wild if the stabilizer limit group is an increasing sequence of stabilizer groups without bound and otherwise is said to be stable if this group chain is bounded. For Cantor actions by a finitely generated group $G$, we prove that stable actions satisfy a rigidity principle and furthermore show that the wild property is an invariant of the continuous orbit equivalence class of the action. A Cantor action is said to be dynamically wild if it is wild and the centralizer limit group is a proper subgroup of the stabilizer limit group. This property is also a conjugacy invariant and we show that a Cantor action with a non-Hausdorff element must be dynamically wild. We then give examples of wild Cantor actions with non-Hausdorff elements, using recursive methods from geometric group theory to define actions on the boundaries of trees.

Publisher

Cambridge University Press (CUP)

Subject

Applied Mathematics,General Mathematics

Cited by 7 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Non-Hausdorff germinal groupoids for actions of countable groups;Indagationes Mathematicae;2024-06

2. Essential holonomy of Cantor actions;Journal of the Mathematical Society of Japan;2024-01-24

3. Prime spectrum and dynamics for nilpotent Cantor actions;Pacific Journal of Mathematics;2023-12-31

4. Settled elements in profinite groups;Advances in Mathematics;2022-08

5. Cantor dynamics of renormalizable groups;Groups, Geometry, and Dynamics;2021-12-06

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