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).