Unusual geodesics in generalizations of Thompson’s group $F$

Abstract

We prove that seesaw words exist in Thompson's group $F(N)$ for $N=2,3,4,\ldots$ 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 method for computing word length. This paper generalizes results by Cleary and Taback, who proved the case $N=2$.