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We study some combinatorial properties of the word metric of Thompson's groupF in the standard two generator finite presentation. We explore connectionsbetween the tree pair diagram representing an element w of F, its normal formin the infinite presentation, its word length, and minimal lengthrepresentatives of it. We estimate word length in terms of the number and typeof carets in the tree pair diagram and show sharpness of those estimates. Inaddition we explore some properties of the Cayley graph of F with respect tothe two generator finite presentation. Namely, we exhibit the form of `deadend' elements in this Cayley graph, and show that it has no `deep pockets'.Finally, we discuss a simple method for constructing minimal lengthrepresentatives for strictly positive or negative words.

Combinatorial properties of Thompson’s group 𝐹