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Topological weak containment

Riley Thornton

TL;DR

We develop a topological analogue of weak containment for group actions, establishing equivalences among local-pattern containment, ultra(co)product constructions, and continuous model theory, with approximate conjugacy matching these notions for Cantor actions. The work shows that in zero-dimensional settings these perspectives align, and that Cantor-space dynamics provide a fertile ground for connecting weak containment to genericity and to geometric group properties. The paper analyzes the existence and nature of top and bottom elements and characterizes antichains, revealing that width can encode ends and other coarse-geometric features of the acting group (e.g., $|\mathbb{R}|$-sized antichains). It concludes with open questions on hyperfiniteness analogues, bottom phenomena for finite-action limits, and the broader landscape of weak containment in topological dynamics.

Abstract

We build on work of Elek and Zucker and develop a topological analogue of the theory of weak containment. We show that definitions in terms of local patterns, containment in ultra(co)products, and continuous model theory are all equivalent, just as in ergodic theory. And, for actions on Cantor space, we show these are all equivalent to approximate conjugacy. Restricting our attention to Cantor space, we connect this theory to questions about generic actions. We show how the shape of the space of weak equivalence classes reflects the geometry of the acting group. And, we show that, for $\mathbb{Z}^2$, there is no smallest limit of finite actions.

Topological weak containment

TL;DR

We develop a topological analogue of weak containment for group actions, establishing equivalences among local-pattern containment, ultra(co)product constructions, and continuous model theory, with approximate conjugacy matching these notions for Cantor actions. The work shows that in zero-dimensional settings these perspectives align, and that Cantor-space dynamics provide a fertile ground for connecting weak containment to genericity and to geometric group properties. The paper analyzes the existence and nature of top and bottom elements and characterizes antichains, revealing that width can encode ends and other coarse-geometric features of the acting group (e.g., -sized antichains). It concludes with open questions on hyperfiniteness analogues, bottom phenomena for finite-action limits, and the broader landscape of weak containment in topological dynamics.

Abstract

We build on work of Elek and Zucker and develop a topological analogue of the theory of weak containment. We show that definitions in terms of local patterns, containment in ultra(co)products, and continuous model theory are all equivalent, just as in ergodic theory. And, for actions on Cantor space, we show these are all equivalent to approximate conjugacy. Restricting our attention to Cantor space, we connect this theory to questions about generic actions. We show how the shape of the space of weak equivalence classes reflects the geometry of the acting group. And, we show that, for , there is no smallest limit of finite actions.

Paper Structure

This paper contains 17 sections, 76 theorems, 151 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Statement of results
  4. Notation
  5. Acknowledgments
  6. Definitions
  7. Local patterns
  8. Gelfand duality
  9. Ultra(co)products
  10. Model theory
  11. The space of actions
  12. Equivalences
  13. Tops, bottoms, and anti-chains
  14. Top actions
  15. Bottom classes
  16. ...and 2 more sections

Key Result

Proposition 1.3

The relation $\preccurlyeq$ is a quasi-order on actions of $\Gamma$.

Figures (1)

  • Figure 1: Any periodic tiling with the same local pattern as the above must have period $2$ mod $3$ in the $x$-direction.

Theorems & Definitions (139)

  • Definition 1.1
  • Definition 1.2
  • Proposition 1.3
  • Proposition 1.4: c.f. Survey
  • Theorem 1.5
  • Theorem 1.6
  • Proposition 1.7
  • Proposition 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 129 more