Bases of the Free Group of Rank 2, Revisited

The free group of rank 2 $\langle X,Y\rangle=F_{2}$ is an important mathematical object, used for example in the theory of words and in Markoff theory. The foundations of the theory were laid down by Nielsen (1918), based on the analysis of the bases of $\langle X,Y\rangle$. We propose a new access to this theory, based on the $(X,Y)$-pairs $(A,B)$, that is elements $A,B\in \langle X,Y\rangle$ with the property that $AB^{-1}A^{-1}B=XY^{-1}X^{-1}Y$. As an application, we give a purely algebraic proof of McShane's identity for one-punctured tori, originally proved in the context of hyperbolic geometry.