- We give applications of the discontinuity function of a discrete group for Fuchsian groups which act on the unit disk and which are finitely generated. We obtain two sets of theorems: One set corresponds to the Euclidean metric. The second set corresponds to the hyperbolic metric. These theorems state inequalities that involve combinatorial quantities (such as counting functions of the elements of the group with a given bounded length, or the order of growth of the group) and geometric quantities (such as distances of the images of a point under a fixed set of generators to the unit circle, or hyperbolic areas of certain disks). This paper is a sequel to the paper “The Discontinuity Function of Discrete Groups and Radius of Schlichtness” by the author, that recently appeared in this journal.