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

Andrew Elvey-Price

Published 2016-11-01Version 1

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). As a result, we show that any Cayley graph property $K$ which satisfies $\text{FFTP}\Rightarrow K\Rightarrow\text{MAC}$ 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. This also shows the new results that Poenaru's condition $P(2)$ and the basepoint loop shortening property both depend on the generating set. We also show that the Cayley graph $\Gamma$ does not have the loop shortening property, so this property also depends on the generating set.