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Thompson's group F is not almost convex

Sean Cleary·Jennifer Taback
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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.

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Sean Cleary, Jennifer Taback. 2002-04-19. Thompson's group F is not almost convex. https://arxiv.org/abs/math/0204249

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