Suppose that $(S,\mathfrak{g})$ is a closed negatively curved Riemannian surface =(or more generally a manifold). Negative curvature guarantees that the non-trivial conjugacy classes $\text{conj}(\pi_1(S))$ are in bijection with the closed geodesics on $(S,\mathfrak{g})$. The marked length spectrum $$ \ell_\mathfrak{g} : \text{conj}(\pi_1(S)) \to \mathbb{R}_{>0} $$ assigns to each conjugacy class the length of the corresponding closed geodesic in $(S,\mathfrak{g})$.
It is a well known fact (due to Dal'bo https://link.springer.com/article/10.1007/BF01235869) that the image of $\ell_\mathfrak{g}$ in $\mathbb{R}$ is not contained in a discrete subgroup $c \mathbb{Z}$ for some $c \in \mathbb{R}$.
My question is: can the image of $\ell_\mathfrak{g}$ be contained in a subgroup of the form $$ c_1 \mathbb{Z} \oplus c_2 \mathbb{Z} \oplus \cdots \oplus c_k \mathbb{Z} = \left\{x \in \mathbb{R}: x = \sum_{i=1}^k n_i c_i \ \text{with} \ n_i \in \mathbb{Z} \right\} $$ for some $c_1,\ldots, c_k \in \mathbb{R}$? That is, can it be contained in a finite rank subgroup?
It seems like a very natural question to ask, but I've not been able to find any results about this. Any help would be greatly appreciated - thanks!
