Group actions with almost normal stabilizers

TL;DR

The paper analyzes minimal actions $(X,G)$ of countable groups on compact spaces where stabilizers concentrate in a fixed almost normal subgroup. It establishes precise equivalences between the existence of finite stabilizers, the almost normal stabilizer structure, and the stabilizer URS/IRS architecture, linking holonomy and lack of essential holonomy to URS/IRS supports. It also shows that actions with finite almost normal stabilizers and no essential holonomy arise as almost finite-to-one factors of essentially free actions, particularly for residually finite groups, and develops a factor-extension framework that preserves no essential holonomy under finite-to-one maps. The work provides explicit partitions, factor constructions, and odometer-like examples to illustrate how almost normal stabilizers shape topological and measure-theoretic dynamics, with implications for URS/IRS realizations and extensions to essentially free dynamics.

Abstract

In this paper, we consider minimal group actions of countable groups on compact Hausdorff spaces by homeomorphisms. We show that the existence of a point with finite stabilizer imposes strong restrictions on the dynamics: the residual set of points then has stabilizers conjugate to the same almost normal subgroup of the acting group. Under certain conditions on the acting group such an action also has no essential holonomy. If the acting group is residually finite, we show that every group action with finite almost normal stabilizers with no essential holonomy arises as an almost finite-to-one factor of an essentially free action.

Theorems & Definitions (71)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Theorem 1.4
• Lemma 1.5
• Theorem 1.6
• Corollary 1.7
• Theorem 1.8
• Proposition 1.9
• Lemma 3.1