# Limit group invariants for non-free Cantor actions

## TL;DR

This paper introduces new invariants for Cantor actions, classifies their stability, and explores wild actions, demonstrating invariance properties and providing examples using geometric group theory techniques.

## Contribution

It defines stabilizer and centralizer limit groups as invariants, establishes stability and wildness criteria, and links wildness to non-Hausdorff elements in Cantor actions.

## Key findings

- Stable actions satisfy a rigidity principle.
- Wildness is an invariant under orbit equivalence.
- Wild actions with non-Hausdorff elements are constructed using geometric group theory.

## 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.