Thompson's group F is not almost convex

TL;DR

This paper proves that Thompson's group F does not satisfy the almost convexity condition by constructing specific element pairs and analyzing their geometric properties, using Fordham's length method and tree pair diagrams.

Contribution

It demonstrates that Thompson's group F fails to meet Cannon's almost convexity condition for any finite n, providing a new geometric insight into its structure.

Findings

Thompson's group F is not almost convex for any n

Constructed element pairs at specific distances to demonstrate non-convexity

Utilized Fordham's length calculation and tree diagram analysis

Abstract

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.