A Cayley graph for F2 × F2 which is not minimally almost convex

We give an example of a Cayley graph [Formula: see text] for the group [Formula: see text] which is not minimally almost convex (MAC). On the other hand, the standard Cayley graph for [Formula: see text] does satisfy the falsification by fellow traveler property (FFTP), which is strictly stronger. As a result, any Cayley graph property [Formula: see text] lying between FFTP and MAC (i.e., [Formula: see text]) 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 [Formula: see text] and the basepoint loop shortening property (LSP) for which dependence on the generating set was previously unknown. We also show that the Cayley graph [Formula: see text] does not have the LSP, so this property also depends on the generating set.

Rational Growth and Almost Convexity of Higher-Dimensional Torus Bundles

Given a matrix $A\in SL(N,\mathbb{Z})$, form the semidirect product $G=\mathbb{Z}^N\rtimes_A \mathbb{Z}$ where the $\mathbb{Z}$-factor acts on $\mathbb{Z}^N$ by $A$. Such a $G$ arises naturally as the fundamental group of an $N$-dimensional torus bundle which fibers over the circle. In this article, we prove that if $A$ has distinct eigenvalues not lying on the unit circle, then there exists a finite index subgroup $H\leq G$ possessing rational growth series for some generating set. In contrast, we show that if $A$ has at least one eigenvalue not lying on the unit circle, then $G$ is not almost convex for any generating set.

A uniform model for almost convexity and rewriting systems

We introduce a topological property for finitely generated groups, called stackable, that implies the existence of an inductive procedure for constructing van Kampen diagrams with respect to a particular finite presentation. We also define algorithmically stackable groups, for which this procedure is an algorithm. This property gives a common model for algorithms arising from both rewriting systems and almost convexity for groups.