Minimal Length Elements of Thompson's Group F

Abstract

Elements of the group are represented by pairs of binary trees and the structure of the trees gives insight into the properties of the elements of the group. The review section presents this representation and reviews the known relationship between elements of F and binary trees. In the main section we give a method of determining the minimal lengths of elements of Thompson's group F in the two generator presentation

$$\left\langle {a,b:\left[ {ab^{ - 1} ,a^{ - 1} ba} \right],\left[ {ab^{ - 1} ,a^{ - 2} ba^2 } \right]} \right\rangle .$$

This method is an effective algorithm in that its order is linear in the size of the trees representing an element of F. We also give a method for constructing all minimal length representatives of an element in F.