There is a very nice way to describe some four-dimensional manifolds called a Lefschetz fibration. It is known that any finitely presented group can be realized as the fundamental group of a Lefschetz fibration. This REU will study relations between groups and the types of Lefschetz fibrations whose fundamental group realize a given group. For example, Lefschetz fibrations have two natural integers associated to them, the genus of the fiber and the number of singular fibers. Given a specific group, what can we say about these two integers?