Fuchsian groups are important for the theory of dessin d'enfants in arithmetic geometry. Wolfart wrote a survey here: http://www.math.uni-frankfurt.de/~wolfart/Artikel/abc.pdf .

One typical application is the following: There is some $c>0$, such that for infinitely many $g$ there are $>g^{c\log g}$ non-isomorphic complex curves $C$ of genus $g$ satisfying $|\mathrm{Aut}(C)|=84(g-1)$. For the proof you connect the number of different curves to the number of normal subgroups of the $(2,3,7)$-triangle group, and then show that Fuchsian groups are virtually surface groups, and have therefore many normal subgroups.

Fuchsian groups are important for the theory of dessin d'enfants in arithmetic geometry. Wolfart wrote a survey here: http://www.math.uni-frankfurt.de/~wolfart/Artikel/abc.pdf .

One typical application is the following: There is some $c>0$, such that for infinitely many $g$ there are $>g^{c\log g}$ non-isomorphic complex curves $C$ of genus $g$ satisfying $|\mathrm{Aut}(C)|=84(g-1)$. For the proof you connect the number of different curves to the number of normal subgroups of the $(2,3,7)$-triangle group, and then show that Fuchsian groups are virtually surface groups, and have therefore many normal subgroups.