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.

The three types of normal sequential effect algebras

A sequential effect algebra (SEA) is an effect algebra equipped with a sequential product operation modeled after the Lüders product (a,b)↦aba on C∗-algebras. A SEA is called normal when it has all suprema of directed sets, and the sequential product interacts suitably with these suprema. The effects on a Hilbert space and the unit interval of a von Neumann or JBW algebra are examples of normal SEAs that are in addition convex, i.e. possess a suitable action of the real unit interval on the algebra. Complete Boolean algebras form normal SEAs too, which are convex only when 0=1.We show that any normal SEA E splits as a direct sum E=Eb⊕Ec⊕Eac of a complete Boolean algebra Eb, a convex normal SEA Ec, and a newly identified type of normal SEA Eac we dub purely almost-convex.Along the way we show, among other things, that a SEA which contains only idempotents must be a Boolean algebra; and we establish a spectral theorem using which we settle for the class of normal SEAs a problem of Gudder regarding the uniqueness of square roots. After establishing our main result, we propose a simple extra axiom for normal SEAs that excludes the seemingly pathological a-convex SEAs. We conclude the paper by a study of SEAs with an associative sequential product. We find that associativity forces normal SEAs satisfying our new axiom to be commutative, shedding light on the question of why the sequential product in quantum theory should be non-associative.

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.