Joint work with James Belk.
Abstract: On the n-ball in a Cayley graph of a finitely presented group, there are two natural metrics: the metric induced by the metric on the whole Cayley graph and the metric of shortest paths that stay inside the ball. If these two metrics coincide, we would call the n-ball convex. We prove that the Cayley graph for Thompsonīs group F has n-balls that are as far from being convex as possible:
For any n, there are two points of distance 2 in the Cayley graph such that any path connecting them within the n-ball has length at least 2n. (Note that any two points in the n-ball can be connected by a path via the identity of length at most 2n. So our result is the strongest possible.)