Paper detail

The topology, geometry and dynamics of free groups. Part I: Outer space, fold paths, and the Nielsen/Whitehead problems

The object of this expository work is to try to unveil the topological/geometric intuition behind the theory of free groups and their automorphism and outer automorphism groups. The method we follow is to focus on a series of problems in the study of free groups, and use the solutions of those problems to motivate topological/geometric tools. We do not aim to write down proofs which minimize the number of alphanumeric characters. We instead strive to write down proofs which maximize the development of broadly applicable geometric tools. In Part I we study problems solved by Nielsen and Whitehead in the 1920's and 1930's, but we approach these problems from a modern topological/geometric viewpoint, and we formulate their solutions so as to motivate modern tools, including marked graphs, the outer space of a free group, and fold paths in outer space.