Unusual Geodesics in generalizations of Thompson's Group F

TL;DR

This paper demonstrates the existence of seesaw words and dead ends in Thompson's Group F(N) for N≥2, revealing complex geodesic structures and non-regularity of geodesic languages in these groups.

Contribution

It extends prior work by proving seesaw words and dead ends exist in all F(N) groups, generalizing results from the case N=2 to all N≥2.

Findings

Seesaw words exist in F(N) for all N≥2.

Dead ends with depth 2 exist in F(N).

F(N) is neither synchronously combable nor has a regular geodesic language.

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 g^k or g^{-k} (for given g\in 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 \Gamma which passes through w can continue past w, and the depth of w is the minimal m\in\mathbb{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.