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

TL;DR

This paper constructs a specific Cayley graph for the group F_2×F_2 that is not minimally almost convex, demonstrating that certain geometric properties of Cayley graphs depend on the choice of generating set.

Contribution

It provides an explicit example of a Cayley graph for F_2×F_2 lacking MAC and explores how various geometric properties depend on generating set choices.

Findings

Constructed a Cayley graph not minimally almost convex.

Showed the standard Cayley graph satisfies FFTP.

Demonstrated property dependence on generating set for several properties.

Abstract

We give an example of a Cayley graph $\Gamma$ 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 Po\'{e}naru'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 $\Gamma$ does not have the loop shortening property, so this property also depends on the generating set.