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A Cayley graph for $F_{2}\times F_{2}$ which is not minimally almost convex

Andrew Elvey Price

Abstract

We give an example of a Cayley graph $Γ$ for the group $F_{2}\times F_{2}$ which is not minimally almost convex (MAC). On the other hand, the standard Cayley graph for $F_{2}\times F_{2}$ does satisfy the falsification by fellow traveler property (FFTP), which is strictly stronger. As a result, any Cayley graph property $K$ lying between FFTP and MAC (i.e., $\text{FFTP}\Rightarrow K\Rightarrow\text{MAC}$) is dependent on the generating set. This includes the well known properties FFTP and almost convexity, which were already known to depend on the generating set as well as Poénaru's condition $P(2)$ and the basepoint loop shortening property for which dependence on the generating set was previously unknown. We also show that the Cayley graph $Γ$ does not have the loop shortening property, so this property also depends on the generating set.

A Cayley graph for $F_{2}\times F_{2}$ which is not minimally almost convex

Abstract

We give an example of a Cayley graph for the group which is not minimally almost convex (MAC). On the other hand, the standard Cayley graph for does satisfy the falsification by fellow traveler property (FFTP), which is strictly stronger. As a result, any Cayley graph property lying between FFTP and MAC (i.e., ) is dependent on the generating set. This includes the well known properties FFTP and almost convexity, which were already known to depend on the generating set as well as Poénaru's condition and the basepoint loop shortening property for which dependence on the generating set was previously unknown. We also show that the Cayley graph does not have the loop shortening property, so this property also depends on the generating set.

Paper Structure

This paper contains 5 sections, 7 theorems, 39 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The path metric and almost convexity conditions
  3. The pair $(G,S_{2})$ does not satisfy MAC
  4. Loop shortening properties
  5. Acknowledgements

Key Result

Lemma 3.2

(Britton's Lemma) Assume $w\in G$ is given by the product where $n\geq 1$, each $p_{j}\in P$ and each $\epsilon_{j}=\pm1$. Assume further that there is no $1\leq j\leq n$ satisfying both $p_{j}\in H$ and $\epsilon_{j}=-\epsilon_{j+1}$. Then $w\neq 1$.

Figures (2)

  • Figure 1: Part of the Cayley graph $\Gamma(G,S_{2})$. The bold vertices and edges are those within the ball of radius 8.
  • Figure 2: The loop $\ell$ shown by a dashed red line and a $k$-fellow travelling loop $\ell'$ shown by a dotted blue line.

Theorems & Definitions (18)

  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • ...and 8 more