Thompson ’ s group F ( n ) is not minimally almost convex

We prove that Thompson’s group F (n) is not minimally almost convex with respect to the standard finite generating set. A group G with Cayley graph Γ is not minimally almost convex if for arbitrarily large values of m there exist elements g, h ∈ Bm such that dΓ(g, h) = 2 and dBm (g, h) = 2m. (Here Bm is the ball of radius m centered at the identity.) We use tree-pair diagrams to represent elements of F (n) and then use Fordham’s metric to calculate geodesic length of elements of F (n). Cleary and Taback have shown that F (2) is not almost convex and Belk and Bux have shown that F (2) is not minimally almost convex; we generalize these results to show that F (n) is not minimally almost convex for all n ∈ {2, 3, 4, . . . }.