Thompsonʹs group F is not almost convex

We show that Thompsonʹs group F does not satisfy Cannonʹs 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 Fordhamʹs method for calculating the length of a word in F and upon an analysis of the generatorsʹ geometric actions on the tree pair diagrams representing elements of F.