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On maximal subgroups of ample groups

Rostislav Grigorchuk, Yaroslav Vorobets

Abstract

The paper is concerned with maximal subgroups of the ample (better known as topological full) groups of homeomorphisms of totally disconnected compact metrizable topological spaces. We describe all maximal subgroups that are stabilizers of finite sets. Under certain assumptions on the ample group (including minimality), we describe all maximal subgroups that are stabilizers of closed sets or stabilizers of partitions into clopen sets. In particular, our results apply to the ample groups associated with Cantor minimal systems.

On maximal subgroups of ample groups

Abstract

The paper is concerned with maximal subgroups of the ample (better known as topological full) groups of homeomorphisms of totally disconnected compact metrizable topological spaces. We describe all maximal subgroups that are stabilizers of finite sets. Under certain assumptions on the ample group (including minimality), we describe all maximal subgroups that are stabilizers of closed sets or stabilizers of partitions into clopen sets. In particular, our results apply to the ample groups associated with Cantor minimal systems.
Paper Structure (9 sections, 73 theorems, 9 equations)

This paper contains 9 sections, 73 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Amplification
  3. Subgroups of a fully amplified group
  4. Stabilizers
  5. Regularity properties
  6. Generalized 2-cycles
  7. Maximal subgroups of ample groups
  8. Nowhere dense closed sets
  9. Property \ref{['prop-E']}

Key Result

Theorem 1.1

Let $\mathcal{G}\subset\mathop{\mathrm{Homeo}}(X)$ be an ample group that acts minimally on $X$. Suppose $H$ is a maximal subgroup of $\mathcal{G}$ that does not act minimally on $X$. Then $H=\mathop{\mathrm{St}}\nolimits_{\mathcal{G}}(Y)$, the stabilizer of some closed set $Y\subset X$ different fr

Theorems & Definitions (152)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Conjecture 1.8
  • Proposition 2.1
  • proof
  • ...and 142 more