Seesaw words in Thompson's group F

TL;DR

This paper introduces a family of words in Thompson's group F that challenge the search for minimal length representatives and demonstrate that F cannot be combed by geodesics, revealing new structural complexities.

Contribution

It identifies specific 'seesaw' words in Thompson's group F that illustrate limitations in existing methods for finding geodesic representatives and show non-combability by geodesics.

Findings

Existence of 'seesaw' words with limited suffix options

Demonstration that F is not geodesically combable

Insights into the structure of geodesics in F

Abstract

We describe a family of words in Thompson's group F which present a challenge to the question of finding canonical minimal length representatives, and which show that F is not combable by geodesics. These words have the property that there are only two possible suffixes of long lengths for geodesic paths to the word from the identity; one is of the form $g^k$ and the other of the form $g^{-k}$ where g is a generator of the group.