Thompson's group F is not almost convex

✍ Scribed by Sean Cleary; Jennifer Taback

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 293 KB
• Volume: 270
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1016/s0021-8693(03)00516-7

✦ Synopsis

We show that Thompson's group F does not satisfy Cannon's almost convexity condition AC(n) for any positive integer n with respect to the standard generating set with two elements. 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 .