In 1911, mathematician Max Dehn posed three decision problems for finitely presented groups that have remained central to the study of combinatorial group theory. His work provided the foundation for geometric group theory, which aims to analyze groups using the topological and geometric properties of the spaces they act on. In this thesis, we study group actions on Cayley graphs and the Farey tree. We prove that a group has a solvable word problem if and only if its associated Cayley graph is constructible. Moreover, we prove that a group is finitely generated if and only if it acts geometrically on a proper path-connected metric space. As an example, we show that SL(2, Z) is finitely generated by proving that it acts geometrically on the Farey tree.