Forest Diagrams and Lengths for the Generalised Thompson's Group $F(n)$

Abstract

We extend the concept of two-way forest diagrams, introduced by Belk and Brown in 2003, to represent elements of $F(n)$ as a pair of infinite, bounded $n$-ary forests together with an order-preserving bijection of the leaves. This representation allows us to develop an alternative way to compute the length of an element of $F(n)$, distinct from the formula established by Fordham and Cleary in 2009. As an application of our length formula, we re-prove the existence of dead end elements in $F(n)$ and show that their depth is always two, first proved by Wladis in 2009.