A Fuchsian model for a Riemann surface $X$ is a discrete subgroup $G$ of $PSL_2(\mathbb{R})$ such that there is a biholomorphic map from $U/G$ to $X$.

For a fixed genus $g \geq 2$ one knows from Bers embedding theorem that there is a holomorphic family of all Fuchsian models parametrized by the Teichmueller space $T_g$. More precisely, the mapping $t \mapsto G_t$ depends holomorphically on $t$ where $G_t$ is the Fuchsian model for the Riemann surface represented by $t$.

My question is: does there exist another parametrization of Fuchsian models which is not the one I described above?

Or another question, that is very much connected to my question: is there another method than Bers' embedding to induce a complex structure on the Teichmueller space $T_g$?