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

TL;DR

This paper generalizes forest diagram representations for the Thompson's group $F(n)$, providing a new length computation method and analyzing dead end elements with fixed depth.

Contribution

It introduces an extended forest diagram approach for $F(n)$ and develops a novel length formula, re-establishing properties of dead end elements.

Findings

New forest diagram representation for $F(n)$

A different length computation method from previous formulas

Dead end elements in $F(n)$ have depth exactly two

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.