[Paper Review] Seesaw words in Thompson's group F

Sean Cleary, Jennifer Taback|arXiv (Cornell University)|Oct 29, 2003

Geometric and Algebraic Topology5 references3 citations

TL;DR

This paper introduces 'seesaw words' in Thompson's group F—elements with only two possible long geodesic suffixes, $x_0^k$ and $x_0^{-k}$—demonstrating that the Cayley graph with standard generators $\{x_0, x_1\}$ fails the $k$-fellow traveler property for any $k$, proving that $F$ is not combable by geodesics. The construction reveals inherent non-uniqueness in minimal length representatives, complicating canonical word reduction.

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.

Motivation & Objective

• To demonstrate that Thompson's group F lacks a canonical minimal length representative for group elements under the standard generating set $\{x_0, x_1\}$.
• To show that the Cayley graph of F with respect to $\{x_0, x_1\}$ fails the $k$-fellow traveler property for any $k$, implying the graph is not combable by geodesics.
• To analyze the geometric complexity of F by constructing elements with highly constrained geodesic paths, revealing structural obstacles to combability.
• To extend understanding of the non-almost convexity and non-combability of F by exhibiting explicit families of elements with divergent geodesic behavior.

Proposed method

• Constructs a family of elements in F, called 'seesaw words', using tree pair diagrams with specific caret type configurations to control word length changes under multiplication by generators.
• Applies Fordham's algorithm for computing word length in F by tracking changes in caret types (e.g., $L_L$, $R_{NI}$) across tree pair diagrams during multiplication.
• Uses the reduction condition on tree pair diagrams to ensure uniqueness and to analyze how $x_0^\pm$ and $x_1^\pm$ affect word length via changes in caret pairing types.
• Analyzes geodesic paths from the identity to seesaw words, showing that only $x_0^k$ or $x_0^{-k}$ can be long suffixes, depending on the path.
• Employs a distance argument between geodesic paths passing through $wx_0$ and $wx_0^{-1}$, showing their $k$-fellow traveler distance grows with swing $k$, violating boundedness.
• Proves that no geodesic combing exists by contradiction: assuming such a combing leads to unbounded fellow traveler distance for large-swing seesaw words.

Experimental results

Research questions

• RQ1 Can canonical minimal length representatives be consistently constructed for elements of Thompson's group F using the standard generating set?
• RQ2 Does the Cayley graph of F with respect to $\{x_0, x_1\}$ satisfy the $k$-fellow traveler property for any constant $k$?
• RQ3 What structural features of F's word metric prevent it from being combable by geodesics?
• RQ4 How do the geometric properties of tree pair diagrams influence word length reduction under generator multiplication?
• RQ5 Are there elements in F for which only two generators can reduce word length, and what implications does this have for geodesic paths?

Key findings

• The paper constructs a family of 'seesaw words' in F where only $x_0^k$ and $x_0^{-k}$ can be long geodesic suffixes, with swing $k$ arbitrarily large.
• For any $k$, there exist seesaw words such that geodesic paths to $w$ must end in either $x_0^k$ or $x_0^{-k}$, and paths through $wx_0$ and $wx_0^{-1}$ have unbounded fellow traveler distance.
• The distance between geodesic paths passing through $wx_0$ and $wx_0^{-1}$ is $2k$, which grows without bound as $k$ increases, violating the $k$-fellow traveler property.
• This implies that Thompson's group F is not combable by geodesics, as no consistent family of geodesic paths from the identity can satisfy the bounded fellow traveler condition.
• The construction shows that minimal length representatives in F are not unique or canonical, as multiple geodesic paths exist to the same element with no bounded distance between them.
• The failure of the $k$-fellow traveler property is demonstrated explicitly via tree pair diagram analysis and word length tracking under generator multiplication.