Thompson's Group F is not Minimally Almost Convex

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.

Thompson's group F is not almost convex

We show that Thompson's group F does not satisfy Cannon's almost convexity condition AC(n) for any integer n in the standard finite two generator presentation. To accomplish this, we construct a family of pairs of elements at distance n from the identity and distance 2 from each other, which are not connected by a path lying inside the n-ball of length less than k for increasingly large k. Our techniques rely upon Fordham's method for calculating the length of a word in F and upon an analysis of the generators' geometric actions on the tree pair diagrams representing elements of F.

一直向下的希格曼 - 汤普森群$F_n$

Higman--Thompson groups $F_n$ all the way down

证明对$n\geq2$，希格曼 - 汤普森群$F_n$有无限指数且同构于自身的极大子群。构造相关子群链，刻画了特定同胚，给出将共轭问题转化为算法的机制，并应用于解决一些群同构问题。

Computing word length in alternate presentations of Thompson's group F

We introduce a new method for computing the word length of an element of Thompson's group F with respect to a "consecutive" generating set of the form X_n={x_0,x_1,...,x_n}, which is a subset of the standard infinite generating set for F. We use this method to show that (F,X_n) is not almost convex, and has pockets of increasing, though bounded, depth dependent on n.

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…

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。