Computing word length in alternate presentations of Thompson's group F

TL;DR

This paper presents a new method for calculating word length in Thompson's group F with respect to a specific generating set, revealing non-almost convexity and pockets of bounded depth.

Contribution

Introduces a novel technique for computing word length in Thompson's group F using a 'consecutive' generating set, and analyzes geometric properties like almost convexity.

Findings

F is not almost convex with respect to the new generating set

Identifies pockets of increasing, bounded depth in (F,X_n)

Provides a practical method for word length computation in F

Abstract

We introduce a new method for computing the word length of an element of Thompson's group F with respect to a "consecutive" generating set of the form X_n={x_0,x_1,...,x_n}, which is a subset of the standard infinite generating set for F. We use this method to show that (F,X_n) is not almost convex, and has pockets of increasing, though bounded, depth dependent on n.