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Limit group invariants for non-free Cantor actions

Part of: Topological dynamics Smooth dynamical systems: general theory

Published online by Cambridge University Press:  09 March 2020

STEVEN HURDER Open the ORCID record for STEVEN HURDER [Opens in a new window]  and
OLGA LUKINA
STEVEN HURDER
Affiliation:
Department of Mathematics, University of Illinois at Chicago, 322 SEO (m/c 249), 851 S. Morgan Street, Chicago, IL 60607-7045, USA email hurder@uic.edu
OLGA LUKINA
Affiliation:
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090Vienna, Austria email olga.lukina@univie.ac.at
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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.

Keywords

minimal Cantor actionscontinuous orbit equivalencerigiditynon-Hausdorff groupoids

MSC classification

Primary: 37B45: Continua theory in dynamics 37C15: Topological and differentiable equivalence, conjugacy, invariants, moduli, classification 37C85: Dynamics of group actions other than ${bf Z}$ and ${bf R}$, and foliations
Secondary: 57S10: Compact groups of homeomorphisms
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 6 , June 2021 , pp. 1751 - 1794
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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