It is well-known that an essential closed curve on a hyperbolic surface (possibly with boundary) is homotopic to a unique closed geodesic. Moreover, if the curve under consideration is simple, then so is the geodesic homotopic to it. A reference is "A primer on mapping class groups" of Farb and Margalit (propositions 1.3 and 1.6). 


It is proved [here](https://arxiv.org/abs/math/0005222) that for a torus with 1 puncture $\Sigma\_{1, 1}$ (endowed with a complete hyperbolic metric) every primitive homology class $h \in H\_1(\Sigma\_{1, 1}, \mathbb{Z})\approx \mathbb{Z}^2$ contains a unique simple closed geodesic. This can be surprising for a beginner like me since the preimage of $h$ under abelianization map $\mathrm{ab}:\pi\_1(\Sigma\_{1, 1})\approx F\_2\rightarrow H\_1(\Sigma\_{1, 1}, \mathbb{Z})$ is infinite. Every homotopy class in this preimage contains a closed geodesic yet only one contains a *simple* closed geodesic. 


My question is: are there examples of hyperbolic surfaces of different topology such that every primitive homology class contains exactly one simple closed geodesic? What are the results/references in this general direction? 



How many simple closed geodesics in a given primitive homology class?
The thrice punctured sphere has no simple closed geodesics. The four-times punctured sphere has a unique simple geodesic in each homology class. In general, it is a result of I. Rivin that the number of simple closed geodesics of length bounded above by $L$ grows like $L^{6g - 6 + 2 c},$ where $c$ is the number of punctures. We can restrict to closed surfaces for simplicity, so $c=0.$ Then, the number of homology classes where there is a simple closed geodesic of length $\leq L$ grows no faster than $L^{2g},$ which means that for uniqueness you have to have $g<2,$ which rules out every hyperbolic surface.


