The loop shortening property and almost convexity

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.

Minimal Almost Convexity

In this article we show that the Baumslag-Solitar group $BS(1,2)$ is minimally almost convex, or $MAC$. We also show that $BS(1,2)$ does not satisfy Poénaru's almost convexity condition $P(2)$, and hence the condition $P(2)$ is strictly stronger than $MAC$. Finally, we show that the groups $BS(1,q)$ for $q \geq 7$ and Stallings' non-$FP_3$ group do not satisfy $MAC$. As a consequence, the condition $MAC$ is not a commensurability invariant.

Tame combing and almost convexity conditions

We give the first examples of groups which admit a tame combing with linear radial tameness function with respect to any choice of finite presentation, but which are not minimally almost convex on a standard generating set. Namely, we explicitly construct such combings for Thompson's group F and the Baumslag-Solitar groups BS(1, p) with p \ge 3. In order to make this construction for Thompson's group F, we significantly expand the understanding of the Cayley complex of this group with respect to the standard finite presentation. In particular we describe a quasigeodesic set of normal forms and…

Convexity properties of Thompson's group F

We prove that Thompson's group F is not minimally almost convex with respect to any generating set which is a subset of the standard infinite generating set for F and which contains x_1. We use this to show that F is not almost convex with respect to any generating set which is a subset of the standard infinite generating set.

一个非极小几乎凸的 $F_{2}\times F_{2}$ 凯莱图

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

本文构造了 $F_{2}\times F_{2}$ 的一个非极小几乎凸凯莱图，证明介于 FFTP 与 MAC 之间的凯莱图性质依赖于生成元集，并首次揭示 Poénaru 条件 $P(2)$、基点回路缩短性质及回路缩短性质同样依赖生成元集。

Rational Growth and Almost Convexity of Higher-Dimensional Torus Bundles

Given a matrix $A\in SL(N,\Z)$, form the semidirect product $G=\Z^N\rtimes_A \Z$ where the $\Z$ factor acts on $\Z^N$ by $A$. Such a $G$ arises naturally as the fundamental group of an $N$-dimensional torus bundle which fibers over the circle. In this paper 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.