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Denote the once punctured torus by $\mathbb{T}$. The fundamental group $\pi_1(\mathbb{T})$ of the once-punctured torus is the free group with two generators.

Now consider a Fuchsian representation $\rho: \pi_1(\mathbb{T}) \to \operatorname{PSL}(2, \mathbb{R})$.

To a choice of two generators $a,b$ of $\pi_1(\mathbb{T})$, we can associate two invariant geodesics $\gamma_a, \gamma_b$ on the upper-half plane $H$ corresponding to $\rho(a), \rho(b)$.

Suppose that $a', b'$ are a different choice of generators for $\pi_1(\mathbb{T})$, how are the invariant geodesics $\gamma_{a'}, \gamma_{b'}$ related (maybe as element of the free homotopy set of $H$) to $\gamma_a, \gamma_b$?

Or, I guess, a better question would be, how are $\rho(a'), \rho(b')$ related to $\rho(a), \rho(b)$ as element of $\operatorname{PSL}(2,\mathbb{R})$?

user7090
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    I am not sure what kind of an answer you are interested in besides the existence of an element of the mapping class group of the torus sending one pair to another pair or the existence of a sequence of Markov moves. (I assume, you are aware of these.) – Moishe Kohan May 13 '21 at 13:53

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