## Illinois Journal of Mathematics

### 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 $N=2$.

#### Article information

Source
Illinois J. Math., Volume 53, Number 2 (2009), 483-514.

Dates
First available in Project Euclid: 23 February 2010

https://projecteuclid.org/euclid.ijm/1266934789

Digital Object Identifier
doi:10.1215/ijm/1266934789

Mathematical Reviews number (MathSciNet)
MR2594640

Zentralblatt MATH identifier
1215.20040

Subjects

#### Citation

Wladis, Claire. Unusual geodesics in generalizations of Thompson’s group $F$. Illinois J. Math. 53 (2009), no. 2, 483--514. doi:10.1215/ijm/1266934789. https://projecteuclid.org/euclid.ijm/1266934789

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