Let $\Gamma=<g_1, \dots, g_n>$ $\subset PGL_2(\mathbb{C})$ be a Schottky group of rank $n$. The group $\Gamma$ is called classical if there exists a set of $2n$ pairwise disjoint closed balls $\{B_1, C_1, \dots, B_n, C_n\}$ such that $g_i(\mathbb{P}^1_{\mathbb{C}} \setminus B_i)= \mathring{C_i}$ and $g_i(\mathbb{P}^1_{\mathbb{C}} \setminus \mathring{B_i})= C_i$ for every $i=1, \dots, n$. The group $\Gamma$ is called iso-classical if the balls $B_i, C_i$ can be taken to be the isometric balls associated to $g_i$ and $g_i^{-1}$ respectively.
It seems to be a difficult open problem to prove that every Riemann surface can be uniformized by a classical Schottky group. Not being an expert of the field, I don't even know if one expects such a statement to be true and what would be the evidence in favor of it.
My question goes in another direction, though: do we know anything at all about those Riemann surfaces that can be uniformized by iso-classical Schottky groups? Do we expect every Riemann surface to fall into this class?
