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Graphical C(3)-T(6) implies CAT(0)

Huaitao Gui

TL;DR

The paper addresses the problem of establishing nonpositive curvature for graphical small cancellation groups, specifically extending the classical $C(3)$-$T(6)$ CAT(0) result to graphical presentations. It analyzes the graphical framework by examining the thickened $X_t$ and non-thickened $X$ complexes, proving that for a graphical $T(q)$ presentation with $Girth(\Gamma)\ge p$, the complex $X$ is locally $CAT(0)$ when $(p,q)=(3,6)$ and locally $CAT(-1)$ when $(p,q)\in\{(4,5),(3,7)\}$, using a detailed study of the link condition and piece lengths (notably that all pieces have length $1$ for $q\ge 5$). The results are extended to the $C(p)$-$T(q)$ setting via subdivisions, and the curvature conclusions yield group-theoretic consequences such as a Tits alternative and fixed-point properties for finitely generated groups acting on the resulting complexes. These curvature and subdivision techniques connect graphical small cancellation to robust geometric group theory, enabling new insights into the dynamics and structure of graphical $C(p)$-$T(q)$ groups.

Abstract

Graphical small cancellation extends the classical small cancellation theory and provides a powerful method for constructing groups with interesting features. In the classical setting, C(3)-T(6) small cancellation complexes are known to admit locally CAT(0) metrics. In this paper, we construct locally CAT(0) metrics for graphical C(3)-T(6) complexes.

Graphical C(3)-T(6) implies CAT(0)

TL;DR

The paper addresses the problem of establishing nonpositive curvature for graphical small cancellation groups, specifically extending the classical - CAT(0) result to graphical presentations. It analyzes the graphical framework by examining the thickened and non-thickened complexes, proving that for a graphical presentation with , the complex is locally when and locally when , using a detailed study of the link condition and piece lengths (notably that all pieces have length for ). The results are extended to the - setting via subdivisions, and the curvature conclusions yield group-theoretic consequences such as a Tits alternative and fixed-point properties for finitely generated groups acting on the resulting complexes. These curvature and subdivision techniques connect graphical small cancellation to robust geometric group theory, enabling new insights into the dynamics and structure of graphical - groups.

Abstract

Graphical small cancellation extends the classical small cancellation theory and provides a powerful method for constructing groups with interesting features. In the classical setting, C(3)-T(6) small cancellation complexes are known to admit locally CAT(0) metrics. In this paper, we construct locally CAT(0) metrics for graphical C(3)-T(6) complexes.
Paper Structure (4 sections, 6 theorems, 2 equations, 6 figures)

This paper contains 4 sections, 6 theorems, 2 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Small cancellation condition $T$
  4. Proof of the results

Key Result

Theorem 1.1

Let $G=\langle f: \Gamma \rightarrow \Theta\rangle$ be a $T(q)$ graphical small cancellation presentation with $Girth(\Gamma)\geq p$, where $Girth(\Gamma)$ denotes the length of a shortest cycle in $\Gamma$, and let $X$ be the non-thickened graphical complex associated to this presentation.

Figures (6)

  • Figure 1: Jasmine diagrams with one to four petals
  • Figure 2: A disk diagram with four petals
  • Figure 3: 2-cycle in $Lk(v,X)$
  • Figure 4: 6-cycle in $Lk(v,X)$
  • Figure 5: Disk diagram with three petals in $X_t$
  • ...and 1 more figures

Theorems & Definitions (15)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Remark 1
  • Definition 1
  • Example 2.1
  • Definition 2
  • Remark 2
  • Lemma 3.1
  • proof
  • ...and 5 more