Fuchsian groups, particularly those which are cocompact, form the tips of several big mathematical icebergs. To put this less metaphorically, several discoveries about Fuchsian groups, obtained using hyperbolic geometry, led to generalizations which fueled the growth of combinatorial and geometric group theory in the 20th century.
By work of Dehn and others, cocompact Fuchsian groups are early examples of geometric computations of isoperimetric functions, aka Dehn functions, which Dehn discovered by working with group invariant polygonal tilings of the hyperbolic plane.
Cocompact Fuchsian groups are also interesting early examples of groups with solvable word and conjugacy problem, using ideas related to Dehn's algorithm.
Cocompact Fuchsian group (together with finite rank free groups) are the early examples of hyperbolic groups in the sense of Gromov, proved using that the group with its word metric is quasi-isometric to the hyperbolic plane.