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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)?

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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 $\mathrm{SL}(2, \mathbb{Z})$.

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