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A local treatment of finite alignment and path groupoids of nonfinitely aligned higher-rank graphs

Malcolm Jones

Abstract

We give a local treatment of finite alignment by identifying the finitely aligned part of any (not necessarily finitely aligned) higher-rank graph. We show the finitely aligned part is itself a constellation and forms a finitely aligned relative category of paths together with the original higher-rank graph. We show that the elements of the finitely aligned part are precisely those whose cylinder sets are compact, which allows us to give novel definitions of locally compact path and boundary-path spaces for nonfinitely aligned higher-rank graphs. We extend a semigroup action and the associated semidirect product groupoid developed by Renault and Williams to define ample Hausdorff path and boundary-path groupoids. The groupoids are amenable for nonfinitely aligned $k$-graphs by a result of Renault and Williams. In the finitely aligned case, the path groupoids coincide with Spielberg's groupoids, and the boundary-path groupoid has an inverse semigroup model via a result of Ortega and Pardo.

A local treatment of finite alignment and path groupoids of nonfinitely aligned higher-rank graphs

Abstract

We give a local treatment of finite alignment by identifying the finitely aligned part of any (not necessarily finitely aligned) higher-rank graph. We show the finitely aligned part is itself a constellation and forms a finitely aligned relative category of paths together with the original higher-rank graph. We show that the elements of the finitely aligned part are precisely those whose cylinder sets are compact, which allows us to give novel definitions of locally compact path and boundary-path spaces for nonfinitely aligned higher-rank graphs. We extend a semigroup action and the associated semidirect product groupoid developed by Renault and Williams to define ample Hausdorff path and boundary-path groupoids. The groupoids are amenable for nonfinitely aligned -graphs by a result of Renault and Williams. In the finitely aligned case, the path groupoids coincide with Spielberg's groupoids, and the boundary-path groupoid has an inverse semigroup model via a result of Ortega and Pardo.

Paper Structure

This paper contains 31 sections, 36 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Topology on the power set of a set
  4. Higher-rank graphs
  5. Groupoids
  6. Semigroup actions
  7. Semidirect product groupoids
  8. A local treatment of finite alignment
  9. Nowhere finitely aligned higher-rank graph
  10. Somewhere finitely aligned higher-rank graph
  11. Properties of the finitely aligned part of a higher-rank graph
  12. The finitely aligned part as a constellation
  13. Lack of a factorisation property
  14. Finitely aligned relative category of paths
  15. Path spaces
  16. ...and 16 more sections

Key Result

Lemma 2.1

Let $\Lambda$ be a countable set. The collection of cylinders in $\mathcal{P}(\Lambda)$ is a basis for a second-countable compact Hausdorff topology on $\mathcal{P}(\Lambda)$. Moreover, each cylinder is compact.

Theorems & Definitions (80)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Example 2.5
  • Lemma 2.6
  • Definition 3.1
  • Example 3.2
  • ...and 70 more