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The Parallel Wall Theorem for CAT(0) even 2-complexes

Carl Kristof-Tessier

TL;DR

The paper proves a 2-dimensional Parallel Wall Theorem for CAT(0) even 2-complexes with finite shapes, extending classical results for Coxeter/Davis complexes. It employs a purely geometric approach based on Alexandrov angles and convex-wall theory, introducing a truncated piecewise Euclidean metric to control local link configurations and reduce the argument to a finite case analysis. The main result establishes a uniform bound $K$ (depending only on $\operatorname{Shapes}(X)$) such that any vertex at combinatorial distance $\ge K$ from a wall is separated from that wall by another wall. This generalizes prior work by Janzen–Wise and Hruska–Wise and advances understanding of wallspace structures in CAT(0) 2-complexes without requiring cocompactness or symmetry.

Abstract

We prove the Parallel Wall Theorem for CAT(0) 2-complexes constructed by regular polygons with an even number of sides. This result extends a combination of the works of Janzen and Wise and Hruska and Wise.

The Parallel Wall Theorem for CAT(0) even 2-complexes

TL;DR

The paper proves a 2-dimensional Parallel Wall Theorem for CAT(0) even 2-complexes with finite shapes, extending classical results for Coxeter/Davis complexes. It employs a purely geometric approach based on Alexandrov angles and convex-wall theory, introducing a truncated piecewise Euclidean metric to control local link configurations and reduce the argument to a finite case analysis. The main result establishes a uniform bound (depending only on ) such that any vertex at combinatorial distance from a wall is separated from that wall by another wall. This generalizes prior work by Janzen–Wise and Hruska–Wise and advances understanding of wallspace structures in CAT(0) 2-complexes without requiring cocompactness or symmetry.

Abstract

We prove the Parallel Wall Theorem for CAT(0) 2-complexes constructed by regular polygons with an even number of sides. This result extends a combination of the works of Janzen and Wise and Hruska and Wise.

Paper Structure

This paper contains 6 sections, 17 theorems, 16 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Angles and convex subspaces of CAT(0) spaces
  3. Even 2-complexes
  4. Walls in even 2-complexes
  5. Truncated piecewise Euclidean structures
  6. The 2-dimensional Parallel Wall Theorem

Key Result

Theorem 1.1

Let $X$ be a $\mathrm{CAT(0)}$ even 2-complex with $\operatorname{Shapes(X)}$ finite. Then there is a bound $K$, depending only on $\operatorname{Shapes}(X)$, satisfying the following: for any wall $\mathcal{W}$ and vertex $v\in X^{(0)}$ at combinatorial distance $\geq K$ from $\mathcal{W}$, there i

Figures (8)

  • Figure 1: $N(\mathcal{W})$ fails to be convex in both the CAT(0) metric and the combinatorial metric.
  • Figure 2: A 6-truncated octagon. The path from $v$ to $w$ passing through the center is geodesic.
  • Figure 3: Proof of \ref{['lem:red_k=2_bounding_squares']}
  • Figure 4: The proof of \ref{['lem:second_cell']}
  • Figure 5: Hexagon case
  • ...and 3 more figures

Theorems & Definitions (35)

  • Theorem 1.1
  • Proposition 2.1: BridsonHaefliger1999
  • Proposition 2.2: BridsonHaefliger1999
  • Lemma 2.3: The disjointness criterion
  • proof
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • Proposition 3.4: BridsonHaefliger1999
  • Definition 3.5
  • ...and 25 more