# Simple curves on hyperbolic tori

In the paper "Simple curves on hyperbolic tori" by McShane and Rivin, they show that if $$T$$ is a hyperbolic once punctured torus, one can define a norm on the homology $$H_1(T,\mathbb{Z})$$ by associating, to every primitive class (nonzero, and not a multiple of another class), the length of the unique simple closed geodesic in that homology class.

In Theorem 3.2, they study the unit ball $$\mathcal{B}_\ell(T)$$ as we vary the hyperbolic structure on $$T$$. In the proof, they say the following: "Pick a simple geodesic $$\gamma$$, and let $$\gamma'$$ be the shortest associated generator of the fundamental group".

In Corollary 1 (right below the theorem), they seem to imply that $$\gamma'$$ is also a geodesic. They call it the "conjugate geodesic". I don't understand what they mean by $$\gamma'$$. Isn't there a unique geodesic in every free homotopy class of curves (equivalently, in every conjugacy class of the fundamental group)?

They do not give the definition of conjugate geodesic, but we can make an educated guess. Suppose that $$F$$ is a free group of rank two. (So, isomorphic to the fundamental group of the once-punctured torus.) We say that a pair of elements $$x$$ and $$y$$ are conjugate primitives if they, together, generate $$F$$. Note that if $$x$$ and $$y$$ are conjugate primitives then so are the following pairs
• $$x$$ and $$xy$$
• $$x$$ and $$yx$$
• $$x$$ and $$xy^{-1}$$
• $$x$$ and $$y^{-1}x$$
• $$xyx$$ and $$xyxxy$$
and so on. (For further details, look up Nielsen automorphisms.) A nice inductive proof shows that conjugate primitive pairs generated this way (up to inner automorphisms) are in bijection with elements of $$\mathrm{SL}(2, \mathbb{Z})$$.