Combinatorial properties of Thompson’s group 𝐹

Abstract

We study some combinatorial consequences of Blake Fordham’s theorems on the word metric of Thompson’s group F F in the standard two generator presentation. We explore connections between the tree pair diagram representing an element w w of F F , its normal form in the infinite presentation, its word length, and minimal length representatives of it. We estimate word length in terms of the number and type of carets in the tree pair diagram and show sharpness of those estimates. In addition we explore some properties of the Cayley graph of F F with respect to the two generator finite presentation. Namely, we exhibit the form of “dead end” elements in this Cayley graph, and show that it has no “deep pockets”. Finally, we discuss a simple method for constructing minimal length representatives for strictly positive or negative words.