Unusual Geodesics in generalizations of Thompson's Group F
Abstract
We prove that seesaw words exist in Thompson's Group F(N) for N=2,3,4,... with respect to the standard finite generating set X. A seesaw word w with swing k has only geodesic representatives ending in gk or g-k (for given g∈ X) and at least one geodesic representative of each type. The existence of seesaw words with arbitrarily large swing guarantees that F(N) is neither synchronously combable nor has a regular language of geodesics. Additionally, we prove that dead ends (or k--pockets) exist in F(N) with respect to X and all have depth 2. A dead end w is a word for which no geodesic path in the Cayley graph which passes through w can continue past w, and the depth of w is the minimal m∈N such that a path of length m+1 exists beginning at w and leaving B|w|. We represent elements of F(N) by tree-pair diagrams so that we can use Fordham's metric. This paper generalizes results by Cleary and Taback, who proved the case N=2.