reference request – \$ PSL_2 ( mathbb {R}) \$ representations of free groups

Let $$S_ {g, n} ^ b$$ denote a gender surface $$g$$ with $$n$$ punctures and $$b$$ boundary components. Suppose $$max {b, n } geq 1$$. It is obvious then that $$S_ {g, n} ^ b$$ the deformation retracts into a bouquet of $$m: = 2g + n + b-1$$ circles and $$pi_1 (S_ {g, n} ^ b)$$ is free on $$m$$ generators.

Let $$m geq 2$$. Yes $$S_ {g, n} ^ b$$ is equipped with a complete hyperbolic metric with a geodesic boundary, then it is known that there is a fuchsian group of the second type $$Gamma$$, acting on the disc, such as $$S_ {g, n} ^ b$$ can be rebuilt as $$PSL_2 ( mathbb {R}) cup (S ^ 1- Lambda ( Gamma)) / Gamma$$ or $$Lambda ( Gamma)$$ is the limit.

My question is this: given a faithful representation $$F_ {2g + n + b-1} to PSL_2 ( mathbb {R})$$, how to find the topological type of the corresponding surface? Notice, for example, $$F_4$$ could describe a surface with the kind $$2$$ and $$1$$ puncture or a kind surface $$1$$ with $$2$$ punctures and $$1$$ limit component.

Ideally, there is a source or article where this type of question has been studied. (Otherwise, I will have to solve it myself).