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 shorted 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)?

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)?

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 associated simple closed geodesic.

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 shorted 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)?

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 shorted 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)?