| In this dissertation we discuss a number of problems on subgroups of free groups by using foldings of graphs. Every finitely generated subgroup of a free group is represented by a labeled directed graph, which is called the core-graph of the subgroup. Firstly, we prove that any reasonable graph can be the core-graph of a subgroup of a free group. Secondly, we farther study the structure of core-graphs and obtain a number of new partial results on the Hanna Neumann conjecture on intersections of finitely generated subgroups of a free group. Finally, once malnormality of subgroups of a free group is characterized in terms of core-graph, we show that almost every set of k reduced words of a free group generates a malnormal subgroup of rank k. |
