Free Groups and Graphs: the Hanna Neumann Theorem

The relationship between free groups and graphs is one of the many examples of the interaction between seemingly foreign branches of mathematics. After defining and proving elementary relations between graphs and groups, this paper will present a result of Hanna Neumann on the intersection of free subgroups. Although Neumann’s orginal upper bound on the rank of the intersection has been improved upon, her conjecture, that the upper bound is half of the one she originally proved, remains an open question.