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 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?