This thesis is an amalgam of the two manuscripts  and  that the author completed during her graduate studies at the University of Chicago. Article  was published in Ergodic Theory and Dynamical Systems and is reprinted with permission here. The first part of the thesis contains results obtained in , where we study Basmajiantype series identities on holomorphic families of Cantor sets associated to one-dimensional complex dynamical systems. We show that the series is absolutely summable if and only if the Hausdorff dimension of the Cantor set is strictly less than one. Throughout the domain of convergence, these identities can be analytically continued and they exhibit nontrivial monodromy. The second part of the thesis follows mainly from . In particular, we prove an inequality that must be satisfied by displacement of generators of free Fuchsian groups, which is the two-dimensional version of the log (2k - 1) Theorem for Kleinian groups due to Anderson-Canary-Culler-Shalen . As applications, we obtain quantitative results on the geometry of hyperbolic surfaces such as the two-dimensional Margulis constant and lengths of a pair of based loops, which improves a result of Buser’s.