Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Higher-rank trees arising from polyhedral graphs

David Pask

TL;DR

The paper develops a constructive framework to produce planar higher-rank trees (rank between $2$ and $4$) from convex polyhedral graphs via Lambek quadrangle clubs and a complete set of coloured commuting squares. It proves that the resulting higher-rank graphs $\Lambda_{E,\mathcal{C}}$ are connected, singly connected, locally convex, acyclic, embed in their fundamental groupoids, and have trivial fundamental groups, making them higher-rank trees. A left-greedy maximal spanning tree algorithm is used with the KPW presentation to certify trivial $\pi_1$, and the work highlights meaningful differences from rank-$1$ trees, including examples of nonplanarity, automorphism phenomena, and potential connections to totally disconnected locally compact simple groups. The approach combines polyhedral geometry, colouring schemes, and gluing techniques to construct and analyze these objects, contributing new explicit higher-rank trees and insight into their topological and algebraic properties.

Abstract

We introduce a new family of higher-rank graphs, whose construction was inspired by the graphical techniques of Lambek \cite{Lambek} and Johnstone \cite{Johnstone} used for monoid and category emedding results. We show that they are planar $k$-trees for $2 \le k \le 4$. We also show that higher-rank trees differ from $1$-trees by giving examples of higher-rank trees having properties which are impossible for $1$-trees. Finally, we collect more examples of higher-rank planar trees which are not in our family.

Higher-rank trees arising from polyhedral graphs

TL;DR

The paper develops a constructive framework to produce planar higher-rank trees (rank between and ) from convex polyhedral graphs via Lambek quadrangle clubs and a complete set of coloured commuting squares. It proves that the resulting higher-rank graphs are connected, singly connected, locally convex, acyclic, embed in their fundamental groupoids, and have trivial fundamental groups, making them higher-rank trees. A left-greedy maximal spanning tree algorithm is used with the KPW presentation to certify trivial , and the work highlights meaningful differences from rank- trees, including examples of nonplanarity, automorphism phenomena, and potential connections to totally disconnected locally compact simple groups. The approach combines polyhedral geometry, colouring schemes, and gluing techniques to construct and analyze these objects, contributing new explicit higher-rank trees and insight into their topological and algebraic properties.

Abstract

We introduce a new family of higher-rank graphs, whose construction was inspired by the graphical techniques of Lambek \cite{Lambek} and Johnstone \cite{Johnstone} used for monoid and category emedding results. We show that they are planar -trees for . We also show that higher-rank trees differ from -trees by giving examples of higher-rank trees having properties which are impossible for -trees. Finally, we collect more examples of higher-rank planar trees which are not in our family.
Paper Structure (15 sections, 12 theorems, 29 equations)

This paper contains 15 sections, 12 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Coloured graphs
  4. Higher-rank graphs
  5. Gluing higher-rank graphs
  6. Quadrangle clubs built from polyhedral graphs
  7. Convex polyhedra and polyhedral graphs
  8. Quadrangle clubs and half-arc Lambek conditions
  9. The higher-rank graph $\Lambda_{E,\mathcal{C}}$ built from the Lambek quadrangle club $E$ defined from a polyhedral graph $\mathfrak{P}=(P,A)$
  10. Colouring the Lambek quadrangle club
  11. The coloured Lambek quadrangle club $(E,c)$
  12. Computing the Fundamental Group of $\Lambda_{(E,\mathcal{C})}$
  13. Left-greedy maximal spanning tree algorithm
  14. Differences from one-dimensional trees
  15. Tailpiece

Key Result

Lemma 2.14

Let $(E,c)$ be a $k$-coloured graph with commuting squares $\mathcal{C}$. Suppose that $\sim_1$ is an equivalence relation on $E^0 \sqcup E^1$ with the following properties: For $e,e' \in E^1$ if $e \sim_1 e'$ then $c(e) = c(e')$, $s(e) \sim_1 s(e')$, $r(e) \sim_1 r(e')$. Then the relation $\sim$ on

Theorems & Definitions (57)

  • Example 2.1
  • Definition 2.2
  • Remark 2.4
  • Definition 2.7: Locally convex, RSY1
  • Definition 2.8
  • Example 2.9
  • Definition 2.10
  • Definition 2.11
  • Remark 2.12
  • Remark 2.13
  • ...and 47 more