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CAT(0) triangle-pentagon complexes

Ioana-Claudia Lazăr

TL;DR

The paper investigates how CAT$(0)$ metric curvature interacts with combinatorial $7$-location in triangle-pentagon complexes by constructing a subdivision $X^{\star}$ of a CAT$(0)$ triangle-pentagon complex $X$. Endowing $X^{\star}$ with a flattened 5-wheel metric $d^{\star}$, the authors prove it is locally CAT$(0)$, $7$-located, and locally $5$-large, with no $3$- or $4$-large vertices; the $5$-large vertices are precisely the pentagon centers, giving a combinatorial girth of at least $7$ around those centers. This yields examples of $7$-located, locally $5$-large groups via geometric actions, linking metric and combinatorial curvature conditions. The results extend the landscape of curvature-adapted discrete complexes and relate to prior work on $7$-location and $5/8$-type conditions, providing a concrete construction that bridges CAT$(0)$ geometry and global combinatorial locality.

Abstract

We show that a certain triangulation of CAT(0) triangle-pentagon complexes is $7$-located and locally $5$-large. Hereby we give examples of $7$-located, locally $5$-large groups.

CAT(0) triangle-pentagon complexes

TL;DR

The paper investigates how CAT metric curvature interacts with combinatorial -location in triangle-pentagon complexes by constructing a subdivision of a CAT triangle-pentagon complex . Endowing with a flattened 5-wheel metric , the authors prove it is locally CAT, -located, and locally -large, with no - or -large vertices; the -large vertices are precisely the pentagon centers, giving a combinatorial girth of at least around those centers. This yields examples of -located, locally -large groups via geometric actions, linking metric and combinatorial curvature conditions. The results extend the landscape of curvature-adapted discrete complexes and relate to prior work on -location and -type conditions, providing a concrete construction that bridges CAT geometry and global combinatorial locality.

Abstract

We show that a certain triangulation of CAT(0) triangle-pentagon complexes is -located and locally -large. Hereby we give examples of -located, locally -large groups.
Paper Structure (11 sections, 8 theorems, 2 figures)

This paper contains 11 sections, 8 theorems, 2 figures.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Acknowledgements:
  4. Preliminaries
  5. Generalities
  6. Triangle-pentagon complexes
  7. CAT(0) spaces
  8. Local $k$-largeness
  9. $7$-location
  10. Geometric group actions
  11. The main result

Key Result

Theorem 1

(Lemma $3.2$, Theorem $3.3$) A CAT(0) triangle-pentagon complex admits a triangulation which, when endowed with a particular metric, is $7$-located and locally $5$-large.

Figures (2)

  • Figure 1: Planar dwheel and nonplanar dwheel
  • Figure 2:

Theorems & Definitions (26)

  • Theorem
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.1
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark
  • ...and 16 more